Permutations, Combinations and The Binomial Theorem Unit 9 Chapter 11 in Text Approximately 7 classes
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
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).
Tree diagram possible number of lunch combos Sandwich Side Drink Chicken Soup Yogurt Fruit Juice Milk Juice Milk Juice Milk
Tree diagram possible number of lunch combos Sandwich Side Drink Turkey Soup Yogurt Fruit Juice Milk Juice Milk Juice Milk
Tree diagram possible number of lunch combos Sandwich Side Drink Soup Juice Milk Grilled Cheese Yogurt Juice Milk Fruit Juice Milk
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
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?
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.
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...
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 3 3 9 possible combos
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? 3 3 2 18 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.
Arrangements with and without restrictions
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.
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.
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?
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?
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?
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: 8 12 4 24
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.
Example 2. How many possible outcomes exist if we first flip a coin and then roll a die? Answer: 2 6 12 possible outcomes Example 3. How many possible outcomes exist if we either flip a coin or roll a die? Answer: 2 + 6 = 8 possible outcomes
Example 4. Determine the number of ways that, on a single die, the result could be odd or greater than 4?
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
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! 6 5 4 3 2 1 720
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)... 3 2 1. Note: n! represents the number of ways to arrange n distinct objects in a row.
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?
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.
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 120 2 60
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! 720 720 3! 3! 6* 6 36 20
Practice. How many ways can you arrange 1. FUNCTION the letters in the words below? 2. CANADA 3. MISSISSIPPI
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?
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?
Factorials (revisted) What is the largest factorial that can be calculated by your: calculator? phone?
Simplify the factorial expressions. 5! A) 4! 5 4! 4! 5 150! B) 149! 5 4 3 2 1 4 3 2 1 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.
1000! C ) 997!
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)!
E ) ( n 1)! n 3!
SOLVE 3!( n 1)! : 72 2! n 2!
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.
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.
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
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 5 4 3 2 1 (6 3)! 3! 3 2 1 different 3-letter words
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 12! (12 3)!
Example 3. In how many ways can you arrange 4 pictures in a row on your desk from a collection of 9 pictures?
Note! On many calculators, there is a permutation feature. On the TI-83, we could solve the previous example as follows: P 9 4 9 Math PRB P enter 4 3024 different photo arrangements n 2
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!
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.
Expand by cancelling factorials 1. P 9 2 2. P 6 4 3. P 3 3
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)!
Express each of the following as a permutation 1. 10! 6! 2. 22! 15! 3. 33! 5!
4. n! ( n 2)! 5. ( n 4)! ( n 1)!
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
2.Solve for n: n P 2 = 132
3. Solve for n: P 20 n 3 2
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.
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.
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?
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?
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.
Homework / Practice Page 525 # 10, 11, 12
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.
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
We could also write this answer using permutations: P P P P 4 1 4 2 4 3 4 4 4 12 24 24 64 different words
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
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 0 + + + + =
Practice. 1. How many one-letter, two-letter, or threeletter words can be formed from the word PENCIL?
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?
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?
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
4. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of 46723819?
5. How many numbers between 999 and 9999 are divisible by 5 and not have repeated digits?
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.
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.
Example 2. How many ways can you arrange the letters in the word COSTUME if all the vowels must be kept together?
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?
Practice. 1. How many way can you order the letters in KEYBOARD if K and B must be kept together?
2. How many ways can the letters in SPIDER be ordered if all the consonants must be kept together?
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?
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?
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.
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! 120 48 72 different ways
Example 2. How many arrangements of the word TREAT are there if no vowels can be together?
Practice. 1. How many ways can you arrange 8 boys and 2 girls in a line if the girls are never together?
2. How many ways can you arrange the letters in the word SCARE if the vowels are never together?
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! 6 6 720 20 different arrangements 36 B
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:
Example 2. How many possible routes can we take from A to B if we can only walk East and South? A B
Example 3. How many possible routes can we take from A to B if we can only walk East and South? A B
B Example 4. How many possible routes are there from A to B if we can only walk West and North? A
Practice : Text pg.525 # 8, 10, 11, 12, 16, 25
Section 11.2. 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.
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.
Determine which examples represent a permutation and which represent a combination. 1. The combination to my locker is 24-15-10 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
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
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
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.
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
B) Complete the following statement: The number of combinations is equal to the number of permutations divided by 2 or 2!
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
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
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
Is this an example of permutations or combinations? Combination How many ways are there to select the three students? 10 ways
C) Complete the following statement: The number of combinations is equal to the number of permutations divided by 6 or 3!
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)!
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?
Note: C C 10 7 10 3 is the same as PROVE: C C n r n n r
Example 2. A) To win the LOTTO 649, a person must choose 6 numbers from 1-49. How many ways can we choose 6 numbers? B) To win the LOTTO MAX, a person must choose 7 numbers from 1-49. How many ways can we choose 7 numbers?
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.
We should also be able to solve problems such as the following, where we apply both combinations and the fundamental counting principle.
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.
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
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?
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?
F) Four Aces? G) 5 cards of the same suit? (Called a FLUSH)
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 12 5 12 P5 12 5 5 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!
Prove: C C n r n n r
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
B) 8 C r = 56
n 1 C ) 20 1
n 1 D) 6 n 1
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Da Last Section 11.3 BINOMIAL THEOREM
Expanding Binomials Expand the following: A) (a + b) 2 B) (a + b) 3
C) (a + b)4
Pascals Triangle:
D) (x + 2) 4 E) (2x -3) 5
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.
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.
You can also determine the coefficients represented in Pascal s triangle using combinations. Complete the next row in each pattern.
x y n Binomial Theorem: 0 1 1 2 2 1 1 0 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
n Binomial Theorem: 0 1 1 2 2 1 1 0 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.
Examples: 1.Find the value of a if the expansion of has 18 terms. 4 7 2 a 5 x
2.Expand: 2 x 3y 4
3.What is the third term in the expansion of : 4 2 a 5 6
4. A) In the expansion of 2 3a b 10, which what is the coefficient of the term containing a 4 b 12
a 3 2 4.B) In the expansion of a in simplified form, contains a 5? 7, which term,
5. Determine the constant term of Express in simplest terms. x x 2 2 6
6. Given that a term in the expansion of ax y 6 5 is 252xy, determine the numerical value of a.
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