10-1. Combinations. Vocabulary. Lesson. Mental Math. able to compute the number of subsets of size r.

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1 Chapter 10 Lesson 10-1 Combinations BIG IDEA With a set of n elements, it is often useful to be able to compute the number of subsets of size r Vocabulary combination number of combinations of n things taken r at a time, n, ( n r ) In earlier lessons, you found answers to counting problems involving permutations Recall that a permutation of r of n items is an ordering of r objects taken from the n objects Numbers of permutations can be found using the Multiplication Counting Principle from Lesson 6-2 In contrast, a combination is a collection of objects in which order does not matter For instance, in making trail mix, if ingredients are added in different orders, the end result is the same The Activity exhibits a process that can be used to determine numbers of combinations Activity Sierra is making trail mix for a hike with friends She has fi ve available ingredients: granola, peanuts, raisins, sunfl ower seeds, and dried blueberries Knowing that her friends have different food preferences, she decides to make small bags of mix, each with a different selection of three ingredients, rather than a big bag with all fi ve ingredients How many bags can she make so that each bag has a different mix? Mental Math In the 26-letter English alphabet, how many different 2-letter initials are possible a if the same letter can be repeated? b if the same letter cannot be repeated? Step 1 Using the Multiplication Counting Principle, Sierra reasons that there are 5 choices for the fi rst ingredient in the mix, 4 choices for the second ingredient, and 3 choices for the third How many mixes does she think she can make? Step 2 Worried that she is going to lose track of what was put in each bag, Sierra makes labels Her label GPB means granola-peanuts-blueberries Make a list of the possibilities Make sure that you avoid duplicate mixes BGP is the same mix as GPB Check your list with another student s list to make sure that you have listed all the possibilities How many did you fi nd? On the trail Trail mix, a nutritious, high energy, and lightweight snack eaten while hiking Step 3 How should Sierra adjust her answer from Step 1 to produce the number of mixes you listed in Step 2? (Hint: If you have chosen three ingredients, say granola-peanuts-blueberries, how many different ways could you write the three-letter label?) Step 4 Repeat Steps 1 3 as if Sierra had six ingredients instead of 5 Check your answer by listing all the possibilities 618 Binomial Distributions

2 Lesson 10-1 A Formula for Counting Combinations Step 1 of the Activity asks for the number of permutations of 3 of 5 things Step 2, by contrast, asks for the number of combinations of 5 things taken 3 at a time Notice the difference between permutations and combinations In permutations, order matters and so different orders are counted as unique In combinations, BGP and GPB are considered the same They are different permutations of 3 letters but they are the same combination of 3 letters QY1 The Activity shows how to compute a number of combinations by dividing two numbers of permutations In Step 1, the number of ways of choosing three ingredients in order is 5 P 3 _ 5! or In Step 3, 2! the number of ways of reordering any set of three ingredients is 3 P 3 3! The number 3! is the number of times each trail mix was counted in Step 1 Thus, to find the number of different unordered mixes, you can divide the number of ordered mixes by the number of times each unordered mix appears _ 5 P (_ 5! 3 _ 2!) _ 5! 3! 3!2! _ P 3 The number of combinations of 5 ingredients taken 3 at a time is written 5 C 3 In general, the number of combinations of n things taken r at a time is written n or ( n r ) Some people read this as n choose r A formula for n can be derived from the formula for n P r by generalizing the argument above QY1 Which situation involves combinations? Which situation involves permutations? a Three scholarships worth $1500, $1000, and $500 are distributed to students from a class of 800 b Three $1000 scholarships are distributed to the students of a class of 800 Theorem (Formula for n ) For all integers n and r, with 0 r n, n _ np r _ n! r! (n - r)!r! Proof Any combination of r objects can be arranged in r! ways So r! n n P r Solve for n n n P r 1_ r! Substitute from the Formula for n P r Theorem Thus, n _ n! (n - r) 1_ r! _ n! (n - r)!r! QY2 Because order does not matter in counting combinations, you can consider n as the number of subsets of size r of a set of n objects GUIDED Example 1 A restaurant menu has ten appetizers and twelve main courses A group of people decides they want to order six appetizers and four main courses In how many ways can they do this? (continued on the next page) QY2 Write the formula for the number of ways of making bags of 4-ingredient trail mix if 9 ingredients are available Do not compute an answer Combinations 619

3 Chapter 10 Solution First consider the appetizers There are 10 appetizers, from which 6 are to be chosen This is like choosing a subset of 6 objects from a set of 10 objects, so use the formula for 10 C 6 10 C 6 _ 10! 4!6! For the main courses, there are 12 dishes, from which? are to be chosen Since all the courses come out at once, use the formula for? C? to get _ 12!? total main course combinations?!?! To compute the total number of meals, use the Multiplication Counting Principle: 210 appetizer combinations? main course combinations? different meals Check Use a calculator QY3 Using Combinations to Count Arrangements As you will see in this chapter, combinations are very important in the calculation of certain probabilities For instance, suppose five pennies are tossed If you want to know the probability of getting three tails, then you need to know how many ways three tails can occur in five tosses To answer this question, think of the pennies as occupying five positions along a line, numbered 1 through QY3 How many different ways are there of ordering 7 main courses from a list of 10? Each way of getting three tails is like picking three of the five numbers For instance, if coins 3, 4, and 5 are tails, as pictured above, then you have picked the numbers 3, 4, and 5 Do you see that this is just like Sierra s problem from the Activity? The number of ways of getting three tails in five tosses is the number of combinations of five things taken three at a time, 5 C 3 QY4 Each time you get exactly three tails in five tosses, the other two coins are heads So the number of ways of getting three tails in five tosses is the same as the number of ways of getting two heads in five tosses That is the idea behind the two solutions to Example 2 QY4 How many ways are there to get 3 heads on a toss of 7 coins? 620 Binomial Distributions

4 Lesson 10-1 Example 2 A vocabulary test has 30 items Each correct answer is worth 1 point and each wrong answer is worth 0 points How many ways are there of scoring 25 points? Solution 1 Think: How many ways are there of selecting 25 questions out of 30 to be correct? 30 C 25 _ 30! 5!25! 142,506 Solution 2 Think: How many ways are there to get exactly 5 out of 30 questions wrong? 30 C 5 _ 30! 25!5! 142,506 Thinking Factorially : Computing Combinations by Hand Sometimes numbers resulting from computing a combination are too large for a calculator s memory or display Even when a calculator can compute and display the exact number of combinations, it can be easier and more illuminating to verify the calculator s results by doing some computations by hand When working with quotients of factorials, common pairs of factors appear and cancel each other The result is a more manageable fraction We describe this method of simplifying calculations as thinking factorially Example 3 A school requires that all participants in sports submit to random drug testing During one round of tests, 25 students are randomly selected from among the 250 students participating in sports What is the probability that the quarterback of the football team and the goalies of both the boys and girls soccer teams are three of the students selected? Solution There are 250 C 25 combinations of 25 players that could be picked This number is huge, about This number is the denominator of the probability to be calculated The numerator is the number of these combinations of 25 players that include the three given players Notice that each such combination includes 22 players from the remaining 247 athletes There is only one way to select the three given players ( 3 C 3 1) and there are 247 C 22 ways to select the other 22 players Thus, the probability that these three players are selected, assuming randomness, is 3 C C C 25 1 _ 247! ( _ ( _ 250! 225!25!) 225!22!) _ 247!25! 250!22! 247! ! ! 22! Combinations 621

5 Chapter 10 Questions COVERING THE IDEAS In 1 4, tell whether the situation uses permutations or combinations You do not have to compute an answer 1 number of ways to assign 25 students to 30 desks 2 number of committees of 5 students out of a class of 30 students 3 number of ways of picking 3 books out of a reading list of 10 books 4 number of different orders in which a family could have five girls out of seven children 5 Match each item on the left with one on the right a combination i an arrangement of objects in which different orderings are distinguishable b permutation ii an arrangement of objects in which different orderings are equivalent 6 a How many combinations of the letters UNESCO, taken five at a time, are possible? b Explain why the result of Part a is the same as the number of combinations taken one at a time c Which would be easier to list, the combinations in Part a or those in Part b? d Explain what would be different if you were asked to count all 5-letter strings from UNESCO The United Nations Educational, In 7 and 8, evaluate 7 ( 28 4 ) 8 34 C 1 9 a How many ways can you flip 10 coins and get exactly 3 heads? b Using the Multiplication Counting Principle, how many total outcomes are possible? c Using your answers from Parts a and b, what is the probability of getting three heads in ten flips? 10 Suppose all 4 members of the school golf team are among the 30 athletes chosen for drug testing at a school with 150 athletes a Write an expression for the number of combinations of players that could be chosen b Write an expression for the number of combinations that include the four golf team members c What is the probability that all 4 golf team members are chosen? APPLYING THE MATHEMATICS In 11 13, evaluate the expression Then explain your answer in terms of choosing objects from a collection 11 ( n 1 ) 12 n C n-1 13 n C n Scientifi c and Cultural Organization (UNESCO) Secretariat Building in Paris, France 622 Binomial Distributions

6 Lesson Pete s A Pizza restaurant menu lists fi fteen different toppings a How many different pizzas could be ordered with six toppings? b How many different pizzas could be ordered with nine toppings? 15 Give an example of a situation that would lead to the number 100 C 10 In 16 18, solve for x 16 x C 5 _ 13P 5 5! 17 7! x C 7 31 P 7 18 ( x x - 2 ) A lottery picks five balls from a bin of 55 white balls labeled 1 to 55 and one ball from a bin of 42 green balls labeled 101 to 142 To win the jackpot, a participant must correctly guess all six numbers, although the order of the white numbers is irrelevant Tickets cost $1 If someone were to buy all possible tickets, how much would it cost? 20 Six points are in a plane with no three of them collinear, as shown a How many triangles can be formed having these points as vertices? b Generalize your result in Part a to the case of n points REVIEW 21 A theater has 17 seats in the first row and there are eight more seats in each subsequent row If the last row has 217 seats, how many rows are there in the theater? (Lesson 8-1) 22 Solve for n: 23! n 21! (Lesson 6-4) 23 When one coin is tossed there are two possible outcomes heads or tails How many outcomes are possible when the following numbers of coins are tossed? (Lesson 6-3) a 2 b 5 c 8 d n In 24 and 25, two bags each contain five slips of paper An angle is drawn on each slip The measures of the angles in each bag are 10, 30, 45, 60, and 90 Let a and b be the measures of the angles pulled from bags 1 and 2, respectively (Lessons 6-1, 4-2) 24 Find P(a + b 90 ) 25 Find P(sin(a + b) sin 90 ) EXPLORATION 26 In professional baseball, basketball, and ice hockey, the champion is determined by two teams playing a first-to-four wins out of seven games series Call the two teams X and Y a In how many different ways can the series occur if team X wins? For instance, two different 6-game series are XXYXYX and XYYXXX (Hint: Determine the number of 4-game series, 5- game series, 6-game series, and 7-game series) b There is an n with n and r both less than 10 which equals the answer for Part a Find n and r c Explain why the combination of Part b answers Part a The Stanley Cup, pictured here, is given to the champion of the National Hockey League after a series like that in Question 26 QY ANSWERS 1 a permutations b combinations 2 9 C 4 _ 9P 4 4! _ 9! 5!4! C 3 35 Combinations 623