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.
2.1 COUNTING PRINCIPLES The counting principle is all about choices you make when given many possibilities.
Draw a tree diagram to determine the number of possible outcomes when we flip 3 coins. List the outcomes.
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 Beef), 1 side (Soup, Yogurt, or Fruit), and 1 drink (Juice or Milk).
TREE DIAGRAM POSSIBLE NUMBER OF LUNCH COMBOS What is the total number of possible lunch combos? 18
IS IT ALWAYS PRACTICAL TO USE A TREE DIAGRAM? What if the lunch combo included dessert (cookie or chips)? There would have to be two more branches added to each possibility. What would be the total number of lunch combos? What would be the total number of lunch combos if we had 7 sandwiches, 8 sides, and 4 drinks? Would we draw a tree diagram for this??
When is the task of listing and counting all of the possible outcomes unrealistic? When there is a large number of choices for each outcome Instead, the Fundamental Counting Principle enables students 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.
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, beef) 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.
EXAMPLE 3. 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?
EXAMPLE 4. IN CANADA, POSTAL CODES CONSIST OF A LETTER-DIGIT-LETTER-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?
PRACTICE/ HOMEWORK: PAGES 73 75 # 2, 7, 9,10, 12, 15
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.
In the case of an or situation, if the tasks are mutually exclusive, they involve two disjoint sets, A and B. n( A B) n( A) n( B)
EXAMPLE 2. IN A STANDARD DECK OF CARDS, HOW MANY WAYS CAN YOU CHOOSE A 3 OR A 7? Using a Venn Diagram, we can see that these events are mutually exclusive. n( A B) n( A) n( B) 4 4 8
In the case of an or situation, if the tasks are NOT mutually exclusive, they involve two NOT disjoint sets, A and B. n( A B) n( A) n( B) n( A B)
EXAMPLE 3. IN A STANDARD DECK OF CARDS, HOW MANY WAYS CAN YOU CHOOSE A DIAMOND OR A 7? Using a Venn Diagram, we can see that these events are NOT mutually exclusive. n( A B) n( A) n( B) n( A B) 13 4 1 16
Example 4. How many possible outcomes exist if we first flip a coin and then roll a die? Answer: 2 6 12 possible outcomes Example 5. How many possible outcomes exist if we either flip a coin or roll a die? Answer: 2 + 6 = 8 possible outcomes
Example 6. Determine the number of way that, on a single die, the result could be odd or greater than 4? Answer: n( A B) n( A) n( B) n( A B) 3 2 1 4
PRACTICE / HOMEWORK Pages 73 75 # 3, 8, 14, 17, 18
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! Example 1. 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
Example 2. How many ways can you arrange five books on a shelf? Example 3. How many ways can you arrange 10 people in a row?
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 letters altogether There are 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 THE LETTERS IN THE WORDS BELOW? 1. FUNCTION 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?
PERMUTATIONS ARRANGING A SUBSET OF ITEMS Sometimes you will be given several items but you only want to arrange a few of them. This is referred to as a Permutation.
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, we did this question by looking at each selection individually: Answer: 6 * 5 * 4 = 120 arrangements
ANOTHER WAY OF DOING THIS QUESTION IS TO USE PERMUTATIONS. The notation for a permutation is n P r n! ( n r)! To answer the previous question, we have 6 letters and we want to choose 3 of them. We write: P 6 3 120 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:
EXAMPLE 3. IN HOW MANY WAYS CAN YOU ARRANGE 4 PICTURES IN A ROW ON YOUR DESK FROM A COLLECTION OF 9 PICTURES?
HOMEWORK / PRACTICE Complete worksheets #324-327
EXPAND AND SIMPLIFY EACH OF THE FOLLOWING WITHOUT USING A CALCULATOR: 1. 5 P 4 2. 8 P 6 3. 3 P 3
REMEMBER THE FOLLOWING! 0! 1 1! 1
CANCELLING WITH FACTORIALS To expand factorials, you multiply all the numbers going down to one. 6! 6 5 4 3 2 1 720 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. 4! Suppose we need to stop at 4 for 6! 6! 6 5 4!
WHY WOULD WE NEED TO STOP A FACTORIAL? Stopping is useful when reducing a fraction or when simplifying a calculation. Example: A) 100! 99! This cannot be done on most calculators because the numbers are too big for the calculator to deal with!
B) 12 11 10 9 8! This can be calculated by typing the complete expression on a calculator OR you could make this into a single factorial and type that in!
EVALUATE: 6! 1. 2. 10 9! 5! 10! 120! 3. 4. 8! 2! 118! 3!
WRITE THE FOLLOWING USING FACTORIAL NOTATION: 5. 4 3 2 1 6. 6 5 4 7. 1000 999 8. 16 15 14 321
EXPAND BY CANCELLING FACTORIALS 9. P 9 2 10. P 6 1 11. P 1000 5
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 4)! ( n 4)( n 3)( n 2)...( 3)( 2)( 1)
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)! 4. n 3 n 2 n 1 n!
PERMUTATIONS REVISITED n! Recall: P n r ( n r)! So 9 P 3 = 9! Now suppose we are given the fraction 6! How can we work backwards to write this as a permutation n P r? What is n? Always the numerator (top) of fraction. How do we find r? Simply subtract denominator from numerator! Top - Bottom
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)!
HOMEWORK: TexT PagEs 81-82 #1-4, 5 a) b) c) d), 6-10
SOLVING EQUATIONS INVOLVING FACTORIALS Example: Find the value of n n 2 n! 8 3!! 9! A) B) 10 C) ( n 1)! 4 3!
D) ( n 1)! n! 6 E) n! ( n 1)! 6
F) n! ( n 2)! 6
SOLVING QUADRATIC FUNCTIONS AND N! To solve a quadratic equation, the two most popular methods are: Factoring Quadratic Formula
FACTORING 1. Removing the GCF: 2 3x 6x 0 3x( x 2) 0 3x 0 or x 2 0 3x 3 0 3 x 0 x 2
2. Difference of Squares: b 2 9 0 ( b 3)( b 3) 0 b 3 or b 3
3. Product Sum Trinomials (Type 1) 2 a 7a 8 equation must 0 2 a 7a 8 0 ( a 8)( a 1) 0 P 8 S 7 { 8, 1} a 8 or a 1
SOLVE: 2 n 4 5n
4. Product Sum Trinomials (Type 2) Decomposition: 2 P 12 2 2n 4n 3n 6 0 2 n( n 2) 3( n 2) 0 ( n 2)(2n 3) 0 n 2 0 n 2 2n 7n 6 0 2n 3 0 2n 3 n 3 2 S 7 {4,3}
SOLVE: 2 2n 15n 7 0
SOLVE: 2 6n 11n 10 0
QUADRATIC FORMULA: If 2 ax bx c 0 Then x b 2 b 4ac 2a
SOLVE: 2 6n 11n 10 0 Using the quadratic formula
PRACTICE: SOLVE FOR X 1. 7x 2 70x 0
2. 2 x 25 0
3. x 2 7x 6 0
4. 2 2x 9x 9 0
WE SOLVE EQUATIONS INVOLVING FACTORIALS IN A SIMILAR WAY. 1. Solve for n: n! ( n 2)! 6 n( n 1)( n 2)! ( n 2)! nn ( 1) 6 n 2 However! Since factorial notation is defined only for natural numbers, expressions like (-2)! or (1/2)! have no meaning n 6 0 ( n 3)( n 2) 0 n 3 or n 2 6
2. Solve for n: ( n 3)! 20 ( n 1)!
Note! You may not always have to solve a quadratic equation. 3. Solve for x: ( n 3)! ( n 4)! 8
4.Solve for n: np2 56
Practice: Text pg. 82 #11 pg. 131 #4
BACK TO COUNTING PROBLEMS
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.
Charlie MUST sit here, so that can only happen in 1 way Example 1. How many ways can Adam, Beth, Charlie, and Doug be seated in a row if Charlie 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 Worksheet page 329
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, or any 2 items from the set, or any 3 items 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 NUMBERS 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 three-letter words can be formed from the word PENCIL?
2. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of 46023819?
3. How many numbers between 999 and 9999 are divisible by 5 and not have repeated digits?
4. How many 4-digit numbers less than 4670 can be formed using the digits 1, 3, 4, 5, 8, 9 if repetitions are not allowed?
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, 5 LEVEL II, AND 5 LEVEL III STUDENTS IN A ROW IF STUDENTS FROM EACH LEVEL MUST STAY TOGETHER?
EXAMPLE 4. HOW MANY WAYS CAN WE ARRANGE THE LETTERS OF THE WORD BANANAS, ASSUMING THE B AND S Hint: There will be duplicates!
PRACTICE. 1. How many way can you order the letters in KEYBOARD if K and D 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, assuming that all of the books are different?
5. How many ways can we arrange the letters of the word CORNER, assuming the vowels 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: 5! 120 When they are always together: 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?
3. How many ways can you arrange the letters A,B,C,D,E,I if the vowels and consonants must alternate?
PATHWAYS AND N!
PATHWAYS AND N! 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
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 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: 6! 20 different paths 3!3!
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
EXAMPLE 4. HOW MANY POSSIBLE ROUTES ARE THERE FROM A TO B IF WE CAN ONLY WALK WEST AND NORTH? B A
EXAMPLE 5. HOW MANY POSSIBLE ROUTES ARE THERE FROM A TO B IF WE CAN ONLY WALK WEST AND NORTH? WHAT ABOUT THE OVERLAP? B A
Practice : Text pg.104-107 # 2, 4, 5, 8, 9, 10, 11, 16, 17,18
SECTION 2.5. 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 SP, CR CR, SA SA, CR SP, SA SA, SP CR, SP CR, SA SP, SA
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? P 5! 5! 2! 60 5 3
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!
2.6. 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)! *** this formula will be given to you on your formula sheet
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 n r key on your calculator. C
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? What are the chances of getting hit by lightning in Canada?
EXAMPLE 2. B) To win LOTTO MAX, a person must choose 7 numbers from 1-49. How many ways can 7 numbers be chosen? What are the chances of getting hit by an asteroid?
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.
EXAMPLE 4. THE ATHLETIC COUNCIL DECIDES TO FORM A SUB-COMMITTEE OF 7 COUNCIL MEMBERS TO LOOK AT HOW FUNDS SHOULD BE SPENT ON SPORTS ACTIVITIES IN THE SCHOOL. THERE ARE 15 MEMBERS 9 MALES AND 6 FEMALES. A) In how many ways can the sub-committee be chosen? B) In how many ways can the committee consist of all males?
C) In how many ways can All of the females be chosen? D) In how many ways can the sub-committee be chosen if Shannon, the volleyball coach, must be included?
EXAMPLE 5. A STANDARD DECK OF 52 CARDS HAS THE FOLLOWING CHARACTERISTICS:
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? C) 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 SALAD? Dorothy writes 12! Ken writes 12 P5 Glenda writes Wanda writes Jimmy writes 12 5 5 P12 12 C7 Who is correct?
2.7. SOLVING COUNTING PROBLEMS In this section, we will look at: Combination Problems with at least, at most, always, never, etc Problems involving both Permutations and Combinations
EXAMPLE 1. THE STUDENT COUNCIL IS FORMING A FINANCE SUB-COMMITTEE OF 5 MEMBERS. THERE ARE 11 COUNCIL MEMBERS, 5 MALES AND 6 FEMALES. HOW MANY DIFFERENT WAYS CAN THE SUB-COMMITTEE CONSIST OF : A) Exactly 3 females? 3 females and 2 males C C 6 3 5 2
B) At least 3 females? 3F2M or 4F1M or 5FzeroM C C C C C 6 3 5 2 6 4 5 1 6 5 This is called direct reasoning
STEPS TO SOLVE COMBINATION PROBLEMS BY DIRECT REASONING: 1. Consider only the cases that reflect the conditions. 2. Determine the number of combinations for each case. 3. ADD the results of step 2 to determine the total number of combinations
C) At least 1 female? 1F4M or 2F3M or 3F2M or 4F1M or 5F0M Note: This problem can also be solved by Indirect Reasoning
STEPS TO SOLVE COMBINATION PROBLEMS BY INDIRECT REASONING: 1. Determine the total number of combinations without any conditions. 2. Consider only cases that do NOT meet the conditions. 3. ADD the results of step 2 to determine the total number of combinations that do NOT meet the conditions. 4. Subtract the results of step 3 from step 1
C) At least 1 female solved by Indirect Reasoning or Total No Females C 11 5 C 5 5 (All males)
EXAMPLE 2. A PLANNING COMMITTEE IS TO BE FORMED FOR A SCHOOL-WIDE EARTH DAY PROGRAM. THERE ARE 17 VOLUNTEERS: 10 STUDENTS AND 7 TEACHERS. HOW MANY WAYS CAN THE PRINCIPAL CHOOSE A 5 PERSON COMMITTEE THAT HAS A LEAST 1 TEACHER?
EXAMPLE 3. USING A STANDARD DECK OF CARDS, HOW MANY DIFFERENT 5-CARD HANDS CAN BE FORMED WITH: A) At most 2 Aces? no Aces or 1 Ace or 2 Aces C C C C C C 4 0 48 5 4 1 48 4 4 2 48 3 2 594 400 ways B) Exactly two pairs of fives and tens? pair of fives and pair of tens and fifth card C C C 4 2 4 2 44 1 1584 ways
C) Exactly two pairs? possible pairs one pair second pair fifth card C 13 2 C 4 2 C 4 2 C 44 1 Answer: C C C C 13 2 4 2 4 2 44 1
d) Three of a kind
EXAMPLE 4. A CLASS CONSISTS OF 5 GIRLS AND 7 BOYS. A COMMITTEE IS TO BE FORMED CONSISTING OF 2 GIRLS AND 3 BOYS? IN HOW MANY WAYS CAN A COMMITTEE BE FORMED IF: A) There are no further restrictions? 2 girls and 3 boys C C 5 2 7 3 B) If the Principal s daughter, Katie, must be on the committee? Katie, other girl, and 3 boys C C C 1 1 4 1 7 3
C) The twins, Peter and Paul, cannot both be on the committee? No restrictions = 350 ways Always together Peter Paul 1 other boy 2 Girls C C C C 1 1 1 1 5 1 5 2 Never together = 350-50 = 300 committees
EXAMPLE 4. GABBY CALCULATED 9C2 TO BE 36 AND 9C7 TO BE 36. A) Explain in words why they are equal. If you choose 2, there are 7 left over. If you choose 7, there are 2 left over. B) Use factorial notation to show they are equal. C C 9 2 9 7 9! ( 9 2)! 2! 9! ( 9 7)! 2! 9! 7! 2! 9! 2! 7!
EXAMPLE 5. COMBINATION PROBLEMS ARE COMMON IN COMPUTER SCIENCE. SUPPOSE THERE IS A SET OF 10 DIFFERENT DATA ITEMS REPRESENTED BY {A, B, C, D, E, F, G, H, I, J} TO BE PLACED INTO FOUR DIFFERENT MEMORY CELLS IN A COMPUTER. ONLY 3 DATA ITEMS ARE TO BE PLACED IN THE FIRST CELL, 4 DATA ITEMS IN THE SECOND CELL, 2 DATA ITEMS IN THE THIRD CELL, AND 1 DATA ITEM IN THE LAST CELL. HOW MANY WAYS CAN THE 10 DATA ITEMS BE PLACED IN THE FOUR MEMORY CELLS? 1 st cell: 3 items out of 10 æ10ö è ç 3 ø = 120 2 nd cell: How many items left? 4 items out of 7 æ 7ö è ç 4ø = 35 3 rd cell: How many items left? 2 items out of 3 æ 3ö è ç 2ø = 3 4 th cell: How many items left? 1 item out of 1 æ1ö è ç1 ø = 1 How do we find the total number of combinations? 120 x 35 x 3 x 1 = 12 600 12 600 combinations
PRACTICE / HOMEWORK Pg. 126-128, #1, 3, 4, 5, 8, 10, 12.
SOLVING FOR N IN COMBINATION PROBLEMS Example 1. Solve for n: A) C n 2 6
B) 21 n C 1 2
C) P 30 n 2 2
D) 3n 4 P2 216
EXAMPLE 2. AFTER A VOLLEYBALL MATCH BETWEEN CBRH AND EVH, THE PLAYERS MET AT THE NET TO SHAKE HANDS. HOW MANY PLAYERS WERE THERE IF THERE WERE 300 HANDSHAKES IN TOTAL? C 300 n 2
PRACTICE / HOMEWORK Pg. 119 #15, Pg. 129 6.