Permutations, Combinations and The Binomial Theorem. Unit 9 Chapter 11 in Text Approximately 7 classes
|
|
- Jeremy Elliott
- 5 years ago
- Views:
Transcription
1 Permutations, Combinations and The Binomial Theorem Unit 9 Chapter 11 in Text Approximately 7 classes
2 In this unit, you will be expected to: Solve problems that involve the fundamental counting principle. Solve problems that involve permutations. Solve problems that involve combinations. The Binomial Theorem
3 Counting Principles The counting principle is all about choices you make when given many possibilities. For example, draw a tree diagram to determine the possible lunch combos if the cafeteria offers a lunch combo for $6 where a person can order 1 sandwich (chicken, turkey, or grilled cheese), 1 side (soup, yogurt, or fruit), and 1 drink (juice or milk).
4 Tree diagram possible number of lunch combos Sandwich Side Drink Chicken Soup Yogurt Fruit Juice Milk Juice Milk Juice Milk
5 Tree diagram possible number of lunch combos Sandwich Side Drink Turkey Soup Yogurt Fruit Juice Milk Juice Milk Juice Milk
6 Tree diagram possible number of lunch combos Sandwich Side Drink Soup Juice Milk Grilled Cheese Yogurt Juice Milk Fruit Juice Milk
7 What is the total number of possible lunch combos? ( c, s, j ),( c, s, m),( c, y, j ),( c, y, m),( c, f, j ),( c, f, m) ( t, s, j ),( t, s, m),( t, y, j ),( t, y, m),( t, f, j ),( t, f, m) ( gc, s, j ),( gc, s, m),( gc, y, j ),( gc, y, m),( gc, f, j ),( gc, f, m) 18
8 Is it always practical to use a tree diagram? What if the lunch combo included dessert (cookie or chips)? What would be the total number of lunch combos? What if we had 7 sandwiches, 8 sides, and 4 drinks?
9 When is the task of listing and counting all of the possible outcomes unrealistic? When the sample size is very large Instead, the Fundamental Counting Principle enables us to find the number of outcomes without listing and counting each one. The Fundamental Counting Principle is the means to find the number of ways of performing two or more operations together. If there are a ways to perform one task and b ways to perform another, then there are a*b ways of performing both.
10 The Fundamental Counting Principle illustrates that multiplying the number of options from each category yields the total possible outcomes. That is, if there are a ways to perform a task, b ways to perform a second task, c ways to perform a third task, etc, then the number of ways of performing all the tasks together is a b c...
11 Example 1. Using the Fundamental Counting Principle, if our lunch combo consists of 1 sandwich (chicken, turkey, grilled cheese) and 1 side (soup, yogurt, fruit), we would have possible combos
12 Example 2. From a previous example, if our lunch combo has 3 sandwiches, 3 sides, and 2 drinks, how many possible combos can we have? possible lunch combos We should realize that we achieved the same number of outcomes as in the tree diagram without having to actually list all the possible outcomes.
13 Arrangements with and without restrictions
14 How many 3 digit numbers are there? This is a problem with no restrictions. How many 3 digit numbers are possible if repeats are not possible? This is a problem with restrictions.
15 In how many ways can five black cars and four red cars be parked next to each other in a parking garage if a black car has to be first and a red car has to be last? Use nine blanks to represent the nine cars parked in a row. There are restrictions. A black car must be in the first position and a red car must be in the last position. Fill these positions first. There are black cars for the first position. There are red cars for the last position. After filling the end positions, there are positions to fill with cars remaining. Use the numbers you have determined to fill in the blanks that represent the nine cars parked in a row.
16 A school cafeteria offers sandwiches made with fillings of ham, salami, cheese, or egg on white, whole wheat, or rye bread. How many different sandwiches can be made using only one filling?
17 In NL a license plate consists of 3 letters followed by 3 numbers. How many license plate arrangements are possible? How many license plate arrangements are possible if no letter or digit can be repeated? How many license plates are possible if vowels (a, e, i, o, u) are not allowed?
18 In Canada, postal codes consist of a letter-digitletter-digit-letter-digit arrangement. How many postal codes are possible? In NL, all postal codes begin with A. How many postal codes are possible in NL?
19 Distinguishing between and and or Example 1. You ve won Student of the Day and you get to pick your prize from 8 CDs, 12 DVDs, or 4 Subway Gift Cards. How many prizes can you select from? Answer:
20 When does the Fundamental Counting Principle apply? The Fundamental Counting Principle applies when tasks are related by the word AND The principle does NOT apply when tasks are related by the word OR.
21 Example 2. How many possible outcomes exist if we first flip a coin and then roll a die? Answer: possible outcomes Example 3. How many possible outcomes exist if we either flip a coin or roll a die? Answer: = 8 possible outcomes
22 Example 4. Determine the number of ways that, on a single die, the result could be odd or greater than 4?
23 Initially, we have 6 letters to pick from. We now have only 5 letters to pick from. 4 letters remaining 3 letters remaining 2 letters remaining 1 letter remaining The Fundamental Counting Principle and N Factorial (n!) How many ways can you arrange the letters in the word MEXICO? Solution: The basic idea is we have 6 objects and 6 possible positions they can occupy. Answer: 6 * 5 * 4 * 3 * 2 * 1 = 720 arrangements
24 There is a shorter way of approaching this problem. It is called the Factorial Function and your calculator will perform this calculation for you! That is: 6!
25 Factorial Notation n! For any positive integer n, n factorial or n! represents the product of all of the positive integers up to and including n. n! = n (n 1) (n 2) Note: n! represents the number of ways to arrange n distinct objects in a row.
26 Example 1. A) How many ways can you arrange five books on a shelf? B) How many ways can you arrange 10 people in a row?
27 C) How many ways can you arrange 1 book on a shelf? One 1! is defined as 1. D) How many ways can you arrange 0 people in a row? ONE 0! is also defined as 1.
28 What happens when we arrange identical items? For example, in the word GABBY, if we rearrange the two Bs, we still get GABBY. Because of this, we have to get rid of the EXTRANEOUS words. We do this, by dividing out the repetitions. To show the different arrangements of the letters in the word GABBY, we write: 5! 2! There are 5 There are letters altogether 2 Bs
29 What happens if we have more than one letter repeat? For example, how many ways can we arrange the letters in the word DEEDED? There are 6 letters altogether There are 3 Ds and 3 Es 6! ! 3! 6*
30 Practice. How many ways can you arrange 1. FUNCTION the letters in the words below? 2. CANADA 3. MISSISSIPPI
31 4. You have 6 balls in a bag 3 red, 2 blue, and 1 white. In how many different ways can you take out the balls if you select one ball at a time and do not replace it in the bag?
32 5.You have 10 Smarties 5 blue, 3 yellow, and 2 pink. In how many different ways can you eat them if you eat one at a time?
33 Factorials (revisted) What is the largest factorial that can be calculated by your: calculator? phone?
34 Simplify the factorial expressions. 5! A) 4! 5 4! 4! 5 150! B) 149! NOTE: If you ever want to stop before reaching 1, you can do so by simply placing the factorial after the number you want to stop at.
35 1000! C ) 997!
36 D) n! n 2! If you want to expand algebraic expressions, you must do so by subtracting one from each term. n! ( n)( n 1)( n 2)...(3)(2)(1) ( n 2)! ( n 2)( n 3)( n 4)...( 3)( 2)( 1) n! ( n)( n 1)( n 2)...(3)(2)(1) n 2! ( n 2)( n 3)( n 4)...(3)(2)(1) OR n! ( n)( n 1)( n 2)! n 2! ( n 2)!
37 E ) ( n 1)! n 3!
38 SOLVE 3!( n 1)! : 72 2! n 2!
39 Permutations arranging a subset of items Sometimes you will be given several items but you only want to arrange a few or ALL of them. This is referred to as a Permutation. A permutation is an ordered arrangement of all or part of a set. For example, the possible permutations of the letters A, B and C are ABC, ACB, BAC, BCA, CAB and CBA. The order of the letters matters.
40 PERMUTATIONS. The notation for a permutation is n P r n! ( n r)! This gives the number permutations of n different (distinct) elements taken r at a time. REMEMBER! Order matters when doing permutations.
41 Initially, we have 6 letters to pick from. We now have only 5 letters to pick from. 4 letters remaining Example 1. How many 3-letter words can be made from the letters A, B, C, D, E, and F? Previously, if we did this question we would look at each selection individually: Answer: 6 * 5 * 4 = 120 arrangements
42 Doing this question using PERMUTATIONS. The notation for a permutation is n! ( n r)! To answer the previous question, we have 6 letters and we want to choose 3 of them. We write: n P r P 6 3 6! 6! (6 3)! 3! different 3-letter words
43 Example 2. In how many ways can you pick a captain, an assistant captain, and a manager from a team with 12 players? Using permutations, we write: P ! (12 3)!
44 Example 3. In how many ways can you arrange 4 pictures in a row on your desk from a collection of 9 pictures?
45 Note! On many calculators, there is a permutation feature. On the TI-83, we could solve the previous example as follows: P Math PRB P enter different photo arrangements n 2
46 When using the formula n r ( n r)! what happens, if r were greater than n? The denominator would contain a factorial of a negative number, which is undefined. P For example: 6 8 (6 8)! 2! This means you are trying to take 6 items and arrange 8 of them at a time. Good luck with that!! P 6! 6! n!
47 When using the formula n! ( n r)! what happens, if r were equal to n? n P r The denominator would contain a factorial of n! n! n! 0, which is so For example: 6! 6! P 6! (6 6)! 0! 6 6 n P n! n ( n n)! (0)! 1 This means you are trying to take 6 items and arrange ALL 6 of them at a time.
48 Expand by cancelling factorials 1. P P P 3 3
49 Don t Forget! You can stop expanding at any term by writing your factorial symbol to indicate the end. Simplify the following expressions: 1. n! ( n 2)! 2. ( n 2)! ( n)! 3 ( n 1)! ( n 3)!
50 Express each of the following as a permutation 1. 10! 6! 2. 22! 15! 3. 33! 5!
51 4. n! ( n 2)! 5. ( n 4)! ( n 1)!
52 Solve equations involving factorials 1. Solve for n: n P 2 = 6 and permutations n! ( n 2)! n( n 1)( n 2)! ( n 2)! nn ( 1) 6 n 2 However! Since factorial notation is defined only for whole numbers, expressions like (-2)! or (1/2)! have no meaning n 6 0 ( n 3)( n 2) 0 n 3 or n 2 6 6
53 2.Solve for n: n P 2 = 132
54 3. Solve for n: P 20 n 3 2
55 4. Solve for n: P 5! 5 r 5! (5 r )! To solve this equation, recall which equals 1 1! = 1 and 0! = 1. Therefore, 1 (5 r)! 5! 5 r = 1 or 5 r = 0, resulting in r = 4, r = 5.
56 Specific Positions Frequently when arranging items, a particular position must be occupied by a particular item. The easiest way to approach these questions is by analyzing how many possible ways each space can be filled.
57 Anri MUST sit here, so that can only happen in 1 way Example 1. How many ways can Anri, Brittany, Cassandra, and Dylan be seated in a row if Anri must be in the second chair?
58 There are 5 consonants There are 2 vowels Example 2. How many ways can you order the letters of KITCHEN if it must start with a consonant and end with a vowel?
59 There are 3 O s and 1 can go here There are 2 O s remaining, and one can go here This position must not be an O, so there are 4 possibilities 4 possibilities here 3 possibilities here 2 possibilities here 1 possibility here Example 3. How many ways can you order the letters of TORONTO if it begins with exactly 2 O s? Careful!!! Don t forget the repetitions! The answer above will need the repetitions divided out.
60 Homework / Practice Page 525 # 10, 11, 12
61 More than one case (Adding) Given a set of items, it is possible to form multiple groups by ordering any 1 item from the set, any 2 items from the set, and so on. If you want the total arrangements from multiple groups, you have to ADD the results of each case.
62 Example 1. How many words (of any number of letters) can be formed from the word math? One-letter words Two-letter words Three-letter words Four-letter words
63 We could also write this answer using permutations: P P P P different words
64 Example 2. How many 3-digit numbers less than 360 can be formed using the digits 1,3,5,7, and 9 if there are no repetitions? The first number is 1. The first number is 3. + = numbers
65 Example 3. How many 3-digit numbers between 499 and 999 are even and have no digits repeat? Ending in 2 Ending in 4 Ending in 6 Ending in 8 Ending in =
66 Practice. 1. How many one-letter, two-letter, or threeletter words can be formed from the word PENCIL?
67 2. How many two-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 A) if repetition is NOT allowed? B) if repetition is allowed?
68 3.How many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6 A) if repetition is NOT allowed? B) if repetition is allowed?
69 How many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6 C) if repetition is NOT allowed and the number is larger than 300 D) if repetition is NOT allowed and the number is even and smaller than 200
70 4. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of ?
71 5. How many numbers between 999 and 9999 are divisible by 5 and not have repeated digits?
72 Always Together: Frequently certain items must always be kept together. To do these questions, you must treat the joined items as if they were only one object.
73 Example 1. How many ways can we arrange 5 boys in a row (Andy, Paul, Mel, Bill, and Dave) if Andy and Paul must always be seated together? We can consider this as 4 items in total Andy Paul Mel Dave Bill that can be arranged 4! ways. Plus, we can arrange Andy and Paul in 2! ways. The total arrangements are : 4!2! = 48 different ways.
74 Example 2. How many ways can you arrange the letters in the word COSTUME if all the vowels must be kept together?
75 Example 3. How many ways can we arrange 5 Level I, 4 Level II, and 6 Level III students in a row if students from each level must stay together?
76 Practice. 1. How many way can you order the letters in KEYBOARD if K and B must be kept together?
77 2. How many ways can the letters in SPIDER be ordered if all the consonants must be kept together?
78 3. How many ways can 4 rock, 5 pop, and 6 classical albums be ordered if all albums of the same genre must be kept together?
79 4. How many ways can 3 math books, 5 chemistry, and 7 physics books be arranged on a shelf if the books of each subject must be kept together?
80 Never Together: As in the previous section, we can figure out when the items are ALWAYS TOGETHER and subtract this number from the TOTAL POSSIBLE ARRANGEMENTS.
81 Example 1. How many ways can we arrange 5 boys in a row (Andy, Paul, Mel, Bill, and Dave) if Andy and Paul must never be seated together? Total number of arrangements: When they are always together: 5! 120 4!2! 48 Therefore, the total number of ways when Andy and Paul are never together are: 5! 4!2! different ways
82 Example 2. How many arrangements of the word TREAT are there if no vowels can be together?
83 Practice. 1. How many ways can you arrange 8 boys and 2 girls in a line if the girls are never together?
84 2. How many ways can you arrange the letters in the word SCARE if the vowels are never together?
85 Pathways and n! A Determining the number of pathways from point A to B is the same as determining the total number of arrangements when there are repetitions. Remember! How many ways can we arrange the letters in the word DEEDED? 6! 720 3! 3! different arrangements 36 B
86 Similarly, we can find the number of possible paths from point A to point B. Example 1. How many ways can we get from Point A to Point B if we can only walk East and South? A 6! 3!3! 20 different paths B From point A to point B we will have to travel 3 blocks East and 3 blocks South. This is the same as writing EEESSS, which means we have:
87 Example 2. How many possible routes can we take from A to B if we can only walk East and South? A B
88 Example 3. How many possible routes can we take from A to B if we can only walk East and South? A B
89 B Example 4. How many possible routes are there from A to B if we can only walk West and North? A
90 Practice : Text pg.525 # 8, 10, 11, 12, 16, 25
91 Section Exploring Combinations In the previous sections, when using the Fundamental Counting Principle or Permutations, the order of items to be arranged mattered. If all you want to do is select items, and don t care what order they re in, you can use combinations.
92 Combination: A grouping of objects where order does NOT matter. For example, the two objects a and b have one combination because ab is the same as ba.
93 Determine which examples represent a permutation and which represent a combination. 1. The combination to my locker is Permutation 2. I like spicy chicken, tomatoes, and mushrooms on my pizza Combination 3. Ann, Bob, Colin, and Debbie are members of the SADD Committee Combination
94 Example 1. As part of the Level III English course, students are required to read the following three books: Catcher in the Rye, A Separate Peace, The Stone Angel A) It is at the teachers discretion as to which order they are taught. List all the different orders in which these three novels can be taught. CR, SP, SA CR, SA, SP SP, CR, SA SP, SA, CR SA, CR, SP SA, SP, CR
95 B) As a student, you are allowed to sign out all three books at the same time. How many different ways can you sign out all three books at the same time? CR, SP, SA
96 Part A) is an example of a permutation where the order is important. Part B) is an example of a combination where the order is NOT important.
97 Example 2. Suppose in example 1, you are only required to read 2 of the books. A) Complete the table to show the number of ways in which teachers and students could do this: Teachers (Permutations) Students (Combinations) CR, SP CR, SP SP, CR CR, SA CR, SA SA, CR SP, SA SP, SA SA, SP
98 B) Complete the following statement: The number of combinations is equal to the number of permutations divided by 2 or 2!
99 Example 3. Five students, Ann, Byron, Chad, Diane, and Ellen take part in a cross country race to represent CBRH. A) Suppose the winner of the race wins $100, the runner-up wins $50 and third place wins $25. The table below shows all the possible ways in which the three prizes could be awarded ABC ACB BAC BCA CAB CBA ABD ADB BAD BDA DAB DBA ABE AEB BAE BEA EAB EBA ACD ADC CAD CDA DAC DCA ACE AEC CAE CEA EAC ECA ADE AED DAE DEA EAD EDA BCD BDC CBD CDB DBC DCB BCE BEC CBE CEB EBC ECB BDE BED DBE DEB EBD EDB CDE CED DCE DEC ECD EDC
100 Is this an example of permutations or combinations? Permutation How many ways are there to award the three prizes? 5 3 5! 2! P 60
101 B) For participating in the cross country race, the school has been awarded three places at a running clinic. The school coach decides to select the 3 lucky students from the 5 students who took part in the cross country race. Using the table from part A), circle the different ways three students can be chosen. ABC ACB BAC BCA CAB CBA ABD ADB BAD BDA DAB DBA ABE AEB BAE BEA EAB EBA ACD ADC CAD CDA DAC DCA ACE AEC CAE CEA EAC ECA ADE AED DAE DEA EAD EDA BCD BDC CBD CDB DBC DCB BCE BEC CBE CEB EBC ECB BDE BED DBE DEB EBD EDB CDE CED DCE DEC ECD EDC
102 Is this an example of permutations or combinations? Combination How many ways are there to select the three students? 10 ways
103 C) Complete the following statement: The number of combinations is equal to the number of permutations divided by 6 or 3!
104 Combinations The number of combinations of n items taken r at a time is given by the formula: n Cr n r n! r!( n r)!
105 Example 1. Three students from a class of ten are to be chosen to go on a school trip. A) In how many ways can they be selected? Write the answer in factorial notation and evaluate. B) Confirm the answer using the key on your calculator. n C r C) In how many ways can students NOT be selected to go on the trip?
106 Note: C C is the same as PROVE: C C n r n n r
107 Example 2. A) To win the LOTTO 649, a person must choose 6 numbers from How many ways can we choose 6 numbers? B) To win the LOTTO MAX, a person must choose 7 numbers from How many ways can we choose 7 numbers?
108 Example 3. Triangles can be formed in an octagon by connecting any 3 of its vertices. Determine the number of different triangles that can be formed in an octagon.
109 We should also be able to solve problems such as the following, where we apply both combinations and the fundamental counting principle.
110 Example 4. A baseball team has 5 pitchers, 6 outfielders and 10 infielders. For a game, the manager needs to field a starting group with 1 pitcher, 3 outfielders and 5 infielders. How many ways can she select the starting group? There are C 5 1, or, ways to select a pitcher. There are C, or, ways to select the outfielders. 6 3 C There are 10 5, or ways to select the infielders. We can apply the fundamental counting principle to determine the ways to select the starting group.
111 Example 5. A standard deck of 52 cards has the following characteristics: 4 suits (Spades, Hearts, Diamonds, and Clubs) Two suits are black (Spades and Clubs) Two suits are red (Hearts and Diamonds) Each suit has 13 cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King) Face cards are the Jacks, Queens, and Kings in each suit
112 Poker is a card game played from a deck of 52 cards A) How many different 5-card poker hands are possible? B) In how many of these 5-card hands will there be all diamonds?
113 C) In how many of these 5-card hands will there be 4 black cards and 1 red card? D) 3 Kings and 2 Aces? E) 3 Kings?
114 F) Four Aces? G) 5 cards of the same suit? (Called a FLUSH)
115 Example 6. How many ways can you choose 5 vegetables out of a possible 12 vegetables to make a tossed salad? Dorothy writes 12! Ken writes Glenda writes Wanda writes Jimmy writes P P12 12 C7 C 12 7 Who is correct? First of all this is a combination. This leaves Glenda and Jimmy. And they are both correct! Selecting 5 vegetables from 12, is the same as NOT selecting 7 vegetables. DO THE MATH!
116 Prove: C C n r n n r
117 Given a value of k, k ε N, solve n C r = k or for either n or r. n k r A) n C 2 =15
118 B) 8 C r = 56
119 n 1 C ) 20 1
120 n 1 D) 6 n 1
121 Page #1-11, 13-15, 17-21, 23, C1
122 Da Last Section 11.3 BINOMIAL THEOREM
123 Expanding Binomials Expand the following: A) (a + b) 2 B) (a + b) 3
124 C) (a + b)4
125 Pascals Triangle:
126 D) (x + 2) 4 E) (2x -3) 5
127 2.a) Use Pascal s triangle to expand (x + y) 7. b) Identify patterns in the expansion of (x + y) 7. i) There are terms in the expansion of (x + y) 7. ii) The powers of x from to in (increase or decrease) successive terms of the expansion. iii) The powers of y from to. (increase or decrease) iv) Each term is of degree (the sum of the exponents for x and y is for each term). v) The coefficients are symmetrical, and they begin with and end with.
128 In the expansion of the binomial (x + y) n, where n N, the coefficients of the terms are identical to the numbers in the (n + 1) th row of Pascal s triangle.
129 You can also determine the coefficients represented in Pascal s triangle using combinations. Complete the next row in each pattern.
130 x y n Binomial Theorem: n n n n n C x y C x y C x y C x y C x y n 0 n 1 n 2 n n 1 n n The binomial theorem is used to expand any power of a binomial, (x + y) n, where n N. Each term in the binomial expansion has the n k k form: C x n k y o where k is the exponent of y and o k + 1 is the term number. Thus, the general term of a binomial n k k expansion is: t C k 1 n k x y Same
131 n Binomial Theorem: x y C x y C x y C n 0 n 1 n 2 x y C n n 1 x y C n n x y n n n n n Important properties of the binomial expansion (x + y) n include the following: The binomial expansions are written in: o descending order of the exponent of the first term in the binomial starting with n and going to 0 o ascending order of the exponent of the second term in the binomial starting with 0 and going to n The expansion contains n + 1 terms. The number of objects, k, selected in the combination n C k can be taken to match the number of factors of the second variable. o That is, it is the same as the exponent on the second variable. The sum of the exponents in any term of the expansion is n.
132 Examples: 1.Find the value of a if the expansion of has 18 terms a 5 x
133 2.Expand: 2 x 3y 4
134 3.What is the third term in the expansion of : 4 2 a 5 6
135 4. A) In the expansion of 2 3a b 10, which what is the coefficient of the term containing a 4 b 12
136 a B) In the expansion of a in simplified form, contains a 5? 7, which term,
137 5. Determine the constant term of Express in simplest terms. x x 2 2 6
138 6. Given that a term in the expansion of ax y 6 5 is 252xy, determine the numerical value of a.
139 Page #5,6,7,10, 12, 14, 15, 19, 20
UNIT 2. Counting Methods
UNIT 2 Counting Methods IN THIS UNIT, YOU WILL BE EXPECTED TO: Solve problems that involve the fundamental counting principle. Solve problems that involve permutations. Solve problems that involve combinations.
More informationPermutations and Combinations
Permutations and Combinations NAME: 1.) There are five people, Abby, Bob, Cathy, Doug, and Edgar, in a room. How many ways can we line up three of them to receive 1 st, 2 nd, and 3 rd place prizes? The
More informationLESSON 4 COMBINATIONS
LESSON 4 COMBINATIONS WARM UP: 1. 4 students are sitting in a row, and we need to select 3 of them. The first student selected will be the president of our class, the 2nd one selected will be the vice
More informationWell, there are 6 possible pairs: AB, AC, AD, BC, BD, and CD. This is the binomial coefficient s job. The answer we want is abbreviated ( 4
2 More Counting 21 Unordered Sets In counting sequences, the ordering of the digits or letters mattered Another common situation is where the order does not matter, for example, if we want to choose a
More informationCHAPTER 8 Additional Probability Topics
CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information
More informationElementary Combinatorics
184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are
More informationMTH 245: Mathematics for Management, Life, and Social Sciences
1/1 MTH 245: Mathematics for Management, Life, and Social Sciences Sections 5.5 and 5.6. Part 1 Permutation and combinations. Further counting techniques 2/1 Given a set of n distinguishable objects. Definition
More informationCS1800: Permutations & Combinations. Professor Kevin Gold
CS1800: Permutations & Combinations Professor Kevin Gold Permutations A permutation is a reordering of something. In the context of counting, we re interested in the number of ways to rearrange some items.
More informationPERMUTATIONS AND COMBINATIONS
PERMUTATIONS AND COMBINATIONS 1. Fundamental Counting Principle Assignment: Workbook: pg. 375 378 #1-14 2. Permutations and Factorial Notation Assignment: Workbook pg. 382-384 #1-13, pg. 526 of text #22
More informationChapter 11, Sets and Counting from Applied Finite Mathematics by Rupinder Sekhon was developed by OpenStax College, licensed by Rice University, and
Chapter 11, Sets and Counting from Applied Finite Mathematics by Rupinder Sekhon was developed by OpenStax College, licensed by Rice University, and is available on the Connexions website. It is used under
More informationUnit 5 Radical Functions & Combinatorics
1 Unit 5 Radical Functions & Combinatorics General Outcome: Develop algebraic and graphical reasoning through the study of relations. Develop algebraic and numeric reasoning that involves combinatorics.
More informationIn this section, we will learn to. 1. Use the Multiplication Principle for Events. Cheesecake Factory. Outback Steakhouse. P.F. Chang s.
Section 10.6 Permutations and Combinations 10-1 10.6 Permutations and Combinations In this section, we will learn to 1. Use the Multiplication Principle for Events. 2. Solve permutation problems. 3. Solve
More informationAdvanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY
Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY 1. Jack and Jill do not like washing dishes. They decide to use a random method to select whose turn it is. They put some red and blue
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More informationUnit 5 Radical Functions & Combinatorics
1 Graph of y Unit 5 Radical Functions & Combinatorics x: Characteristics: Ex) Use your knowledge of the graph of y x and transformations to sketch the graph of each of the following. a) y x 5 3 b) f (
More informationIntroduction. Firstly however we must look at the Fundamental Principle of Counting (sometimes referred to as the multiplication rule) which states:
Worksheet 4.11 Counting Section 1 Introduction When looking at situations involving counting it is often not practical to count things individually. Instead techniques have been developed to help us count
More informationWelcome to Introduction to Probability and Statistics Spring
Welcome to 18.05 Introduction to Probability and Statistics Spring 2018 http://xkcd.com/904/ Staff David Vogan dav@math.mit.edu, office hours Sunday 2 4 in 2-355 Nicholas Triantafillou ngtriant@mit.edu,
More informationCounting Things. Tom Davis March 17, 2006
Counting Things Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles March 17, 2006 Abstract We present here various strategies for counting things. Usually, the things are patterns, or
More informationExercises Exercises. 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}?
Exercises Exercises 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}? 3. How many permutations of {a, b, c, d, e, f, g} end with
More informationTImath.com. Statistics. Too Many Choices!
Too Many Choices! ID: 11762 Time required 40 minutes Activity Overview In this activity, students will investigate the fundamental counting principle, permutations, and combinations. They will find the
More informationChapter 10A. a) How many labels for Product A are required? Solution: ABC ACB BCA BAC CAB CBA. There are 6 different possible labels.
Chapter 10A The Addition rule: If there are n ways of performing operation A and m ways of performing operation B, then there are n + m ways of performing A or B. Note: In this case or means to add. Eg.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MATH 00 -- PRACTICE EXAM 3 Millersville University, Fall 008 Ron Umble, Instr. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given question,
More informationChapter 2 Math
Chapter 2 Math 3201 1 Chapter 2: Counting Methods: Solving problems that involve the Fundamental Counting Principle Understanding and simplifying expressions involving factorial notation Solving problems
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.-9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
More informationDiscrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
More informationMAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
More informationSec. 4.2: Introducing Permutations and Factorial notation
Sec. 4.2: Introducing Permutations and Factorial notation Permutations: The # of ways distinguishable objects can be arranged, where the order of the objects is important! **An arrangement of objects in
More informationUnit on Permutations and Combinations (Counting Techniques)
Page 1 of 15 (Edit by Y.M. LIU) Page 2 of 15 (Edit by Y.M. LIU) Unit on Permutations and Combinations (Counting Techniques) e.g. How many different license plates can be made that consist of three digits
More informationCISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest
More information6. In how many different ways can you answer 10 multiple-choice questions if each question has five choices?
Pre-Calculus Section 4.1 Multiplication, Addition, and Complement 1. Evaluate each of the following: a. 5! b. 6! c. 7! d. 0! 2. Evaluate each of the following: a. 10! b. 20! 9! 18! 3. In how many different
More informationPermutations and Combinations. Quantitative Aptitude & Business Statistics
Permutations and Combinations Statistics The Fundamental Principle of If there are Multiplication n 1 ways of doing one operation, n 2 ways of doing a second operation, n 3 ways of doing a third operation,
More informationchapter 2 COMBINATORICS 2.1 Basic Counting Techniques The Rule of Products GOALS WHAT IS COMBINATORICS?
chapter 2 COMBINATORICS GOALS Throughout this book we will be counting things. In this chapter we will outline some of the tools that will help us count. Counting occurs not only in highly sophisticated
More informationEECS 203 Spring 2016 Lecture 15 Page 1 of 6
EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More information11.3B Warmup. 1. Expand: 2x. 2. Express the expansion of 2x. using combinations. 3. Simplify: a 2b a 2b
11.3 Warmup 1. Expand: 2x y 4 2. Express the expansion of 2x y 4 using combinations. 3 3 3. Simplify: a 2b a 2b 4. How many terms are there in the expansion of 2x y 15? 5. What would the 10 th term in
More informationTriangle Similarity Bundle
Triangle Similarity Bundle 2012/2014Caryn White 1 Triangle Similarity Bundle By Caryn White Table of Contents Triangle Similarity Bundle... 2 Copy Right Informations:... 3 Triangle Similarity Foldable...
More information12.1 The Fundamental Counting Principle and Permutations
12.1 The Fundamental Counting Principle and Permutations The Fundamental Counting Principle Two Events: If one event can occur in ways and another event can occur in ways then the number of ways both events
More informationBlock 1 - Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
More informationNwheatleyschaller s The Next Step...Conditional Probability
CK-12 FOUNDATION Nwheatleyschaller s The Next Step...Conditional Probability Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) Meery To access a customizable version of
More informationDate Topic Notes Questions 4-8
These Combinatorics NOTES Belong to: Date Topic Notes Questions 1. Chapter Summary 2,3 2. Fundamental Counting Principle 4-8 3. Permutations 9-13 4. Permutations 14-17 5. Combinations 18-22 6. Combinations
More informationObjectives: Permutations. Fundamental Counting Principle. Fundamental Counting Principle. Fundamental Counting Principle
and Objectives:! apply fundamental counting principle! compute permutations! compute combinations HL2 Math - Santowski! distinguish permutations vs combinations can be used determine the number of possible
More informationName: 1. Match the word with the definition (1 point each - no partial credit!)
Chapter 12 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. SHOW ALL YOUR WORK!!! Remember
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationFundamental Counting Principle
11 1 Permutations and Combinations You just bought three pairs of pants and two shirts. How many different outfits can you make with these items? Using a tree diagram, you can see that you can make six
More informationHonors Precalculus Chapter 9 Summary Basic Combinatorics
Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each
More informationSTATISTICAL COUNTING TECHNIQUES
STATISTICAL COUNTING TECHNIQUES I. Counting Principle The counting principle states that if there are n 1 ways of performing the first experiment, n 2 ways of performing the second experiment, n 3 ways
More informationFinite Mathematics MAT 141: Chapter 8 Notes
Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication
More informationLEVEL I. 3. In how many ways 4 identical white balls and 6 identical black balls be arranged in a row so that no two white balls are together?
LEVEL I 1. Three numbers are chosen from 1,, 3..., n. In how many ways can the numbers be chosen such that either maximum of these numbers is s or minimum of these numbers is r (r < s)?. Six candidates
More informationFundamental Counting Principle
Lesson 88 Probability with Combinatorics HL2 Math - Santowski Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more
More informationThe Fundamental Counting Principle & Permutations
The Fundamental Counting Principle & Permutations POD: You have 7 boxes and 10 balls. You put the balls into the boxes. How many boxes have more than one ball? Why do you use a fundamental counting principal?
More informationName: Class: Date: ID: A
Class: Date: Chapter 0 review. A lunch menu consists of different kinds of sandwiches, different kinds of soup, and 6 different drinks. How many choices are there for ordering a sandwich, a bowl of soup,
More informationPermutations (Part A)
Permutations (Part A) A permutation problem involves counting the number of ways to select some objects out of a group. 1 There are THREE requirements for a permutation. 2 Permutation Requirements 1. The
More informationPermutations and Combinations Practice Test
Name: Class: Date: Permutations and Combinations Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Suppose that license plates in the fictional
More informationAlgebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations
Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)
More informationName: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11
Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value
More informationApril 10, ex) Draw a tree diagram of this situation.
April 10, 2014 12-1 Fundamental Counting Principle & Multiplying Probabilities 1. Outcome - the result of a single trial. 2. Sample Space - the set of all possible outcomes 3. Independent Events - when
More informationCHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY
CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many real-world fields, such as insurance, medical research, law enforcement, and political science. Objectives:
More information4.1. Counting Principles. Investigate the Math
4.1 Counting Principles YOU WILL NEED calculator standard deck of playing cards EXPLORE Suppose you roll a standard red die and a standard blue die at the same time. Describe the sample space for this
More informationPrinciples of Mathematics 12: Explained!
www.math12.com 284 Lesson 2, Part One: Basic Combinations Basic combinations: In the previous lesson, when using the fundamental counting principal or permutations, the order of items to be arranged mattered.
More informationAlgebra II- Chapter 12- Test Review
Sections: Counting Principle Permutations Combinations Probability Name Choose the letter of the term that best matches each statement or phrase. 1. An illustration used to show the total number of A.
More informationFinite Math - Fall 2016
Finite Math - Fall 206 Lecture Notes - /28/206 Section 7.4 - Permutations and Combinations There are often situations in which we have to multiply many consecutive numbers together, for example, in examples
More informationATHS FC Math Department Al Ain Remedial worksheet. Lesson 10.4 (Ellipses)
ATHS FC Math Department Al Ain Remedial worksheet Section Name ID Date Lesson Marks Lesson 10.4 (Ellipses) 10.4, 10.5, 0.4, 0.5 and 0.6 Intervention Plan Page 1 of 19 Gr 12 core c 2 = a 2 b 2 Question
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
More informationMath 3201 Notes Chapter 2: Counting Methods
Learning oals: See p. 63 text. Math 30 Notes Chapter : Counting Methods. Counting Principles ( classes) Outcomes:. Define the sample space. P. 66. Find the sample space by drawing a graphic organizer such
More informationCounting (Enumerative Combinatorics) X. Zhang, Fordham Univ.
Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 Chance of winning?! What s the chances of winning New York Megamillion Jackpot!! just pick 5 numbers from 1 to 56, plus a mega ball number
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More informationFundamental Counting Principle 2.1 Page 66 [And = *, Or = +]
Math 3201 Assignment 2 Unit 2 Counting Methods Name: Fundamental Counting Principle 2.1 Page 66 [And = *, Or = +] Identify the choice that best completes the statement or answers the question. Show all
More informationSection The Multiplication Principle and Permutations
Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
More informationCombinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
More informationMATH CIRCLE, 10/13/2018
MATH CIRCLE, 10/13/2018 LARGE SOLUTIONS 1. Write out row 8 of Pascal s triangle. Solution. 1 8 28 56 70 56 28 8 1. 2. Write out all the different ways you can choose three letters from the set {a, b, c,
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More informationBusiness Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,
More informationMore Probability: Poker Hands and some issues in Counting
More Probability: Poker Hands and some issues in Counting Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationCh. 12 Permutations, Combinations, Probability
Alg 3(11) 1 Counting the possibilities Permutations, Combinations, Probability 1. The international club is planning a trip to Australia and wants to visit Sydney, Melbourne, Brisbane and Alice Springs.
More information2. How many even 4 digit numbers can be made using 0, 2, 3, 5, 6, 9 if no repeats are allowed?
Math 30-1 Combinatorics Practice Test 1. A meal combo consists of a choice of 5 beverages, main dishes, and side orders. The number of different meals that are available if you have one of each is A. 15
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationRecommended problems from textbook
Recommended problems from textbook Section 9-1 Two dice are rolled, one white and one gray. Find the probability of each of these events. 1. The total is 10. 2. The total is at least 10. 3. The total is
More informationCounting and Probability
0838 ch0_p639-693 0//007 0:3 PM Page 633 CHAPTER 0 Counting and Probability The design below is like a seed puff of a dandelion just before it is dispersed by the wind. The design shows the outcomes from
More informationPurpose of Section To introduce some basic tools of counting, such as the multiplication principle, permutations and combinations.
1 Section 2.3 Purpose of Section To introduce some basic tools of counting, such as the multiplication principle, permutations and combinations. Introduction If someone asks you a question that starts
More informationStrings. A string is a list of symbols in a particular order.
Ihor Stasyuk Strings A string is a list of symbols in a particular order. Strings A string is a list of symbols in a particular order. Examples: 1 3 0 4 1-12 is a string of integers. X Q R A X P T is a
More informationCounting Principles Review
Counting Principles Review 1. A magazine poll sampling 100 people gives that following results: 17 read magazine A 18 read magazine B 14 read magazine C 8 read magazines A and B 7 read magazines A and
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationGrade 7/8 Math Circles November 8 & 9, Combinatorial Counting
Faculty of Mathematics Waterloo, Ontario NL G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles November 8 & 9, 016 Combinatorial Counting Learning How to Count (In a New Way!)
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Lecture Notes Counting 101 Note to improve the readability of these lecture notes, we will assume that multiplication takes precedence over division, i.e. A / B*C
More informationSection : Combinations and Permutations
Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
More informationCounting integral solutions
Thought exercise 2.2 20 Counting integral solutions Question: How many non-negative integer solutions are there of x 1 +x 2 +x 3 +x 4 = 10? Thought exercise 2.2 20 Counting integral solutions Question:
More informationName: Spring P. Walston/A. Moore. Topic worksheet # assigned #completed Teacher s Signature Tree Diagrams FCP
Name: Spring 2016 P. Walston/A. Moore Topic worksheet # assigned #completed Teacher s Signature Tree Diagrams 1-0 13 FCP 1-1 16 Combinations/ Permutations Factorials 1-2 22 1-3 20 Intro to Probability
More information19.2 Permutations and Probability Combinations and Probability.
19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities of compound events and solve problems. When are permutations useful in calculating
More informationCIS 2033 Lecture 6, Spring 2017
CIS 2033 Lecture 6, Spring 2017 Instructor: David Dobor February 2, 2017 In this lecture, we introduce the basic principle of counting, use it to count subsets, permutations, combinations, and partitions,
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationFundamental Counting Principle 2.1 Page 66 [And = *, Or = +]
Math 3201 Assignment 1 of 1 Unit 2 Counting Methods Name: Fundamental Counting Principle 2.1 Page 66 [And = *, Or = +] Identify the choice that best completes the statement or answers the question. 1.
More informationCourse Learning Outcomes for Unit V
UNIT V STUDY GUIDE Counting Reading Assignment See information below. Key Terms 1. Combination 2. Fundamental counting principle 3. Listing 4. Permutation 5. Tree diagrams Course Learning Outcomes for
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 19 February 2014 Poker II 19 February 2014 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More information6/24/14. The Poker Manipulation. The Counting Principle. MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical experiments
The Poker Manipulation Unit 5 Probability 6/24/14 Algebra 1 Ins1tute 1 6/24/14 Algebra 1 Ins1tute 2 MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability models MAFS. 7.SP.3:
More informationAlgebra II Probability and Statistics
Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Slide 3 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional Probability
More information