Math in Our World Section 11.2 Combinations

Learning Objectives Distinguish between combinations and permutations. Find the number of combinations of n objects taken r at a time. Use the combination rule in conjunction with the fundamental counting principle.

Combinations A selection of objects without regard to order is called a combination.

EXAMPLE 1 Comparing Permutations and Combinations Given four housemates, Ruth, Elaine, Ama, and Jasmine, list the permutations and combinations when you are selecting two of them. SOLUTION We’ll start with permutations, then eliminate those that have the same two people listed.

EXAMPLE 1 Comparing Permutations and Combinations SOLUTION Permutations Ruth Elaine Elaine Ruth Ruth Ama Ama Ruth Ruth Jasmine Jasmine Ruth Elaine Ama Ama Elaine Elaine Jasmine Jasmine Elaine Ama Jasmine Jasmine Ama Combinations Ruth Elaine Ruth Ama Ruth Jasmine Elaine Ama Elaine Jasmine Ama Jasmine There are 12 permutations, but only 6 combinations.

EXAMPLE 2 Identifying Permutations and Combinations Decide if each selection is a permutation or a combination. (a) From a class of 25 students, a group of 5 is chosen to give a presentation. (b) A starting pitcher and catcher are picked from a 12-person intramural softball team. SOLUTION (a) This is a combination because there are no distinct roles for the 5 group members, so order is not important. (b) This is a permutation because each selected person has a distinct position, so order matters.

The Combination Rule The number of combinations of n objects taken r at a time is denoted by nCr , and is given by the formula

EXAMPLE 3 Using the Combination Rule How many combinations of four objects are there taken two at a time? SOLUTION Since this is a combination problem, the answer is This matches our result from Example 1.

EXAMPLE 4 An Application of Combinations While studying abroad one semester, Tran is required to visit 10 different cities. He plans to visit 3 of the 10 over a long weekend. How many different ways can he choose the 3 to visit? Assume that distance is not a factor. SOLUTION The problem doesn’t say anything about the order in which they’ll be visited, so this is a combination problem.

EXAMPLE 5 Choosing a Committee At one school, the student government consists of seven women and five men. How many different committees can be chosen with three women and two men? SOLUTION First, we will choose three women from the seven candidates. This can be done in 7C3 = 35 ways. Then we will choose two men from the five candidates in 5C2 = 10 ways. Using the fundamental counting principle, there are 35 x 10 = 350 possible committees.

EXAMPLE 6 Designing a Calendar To raise money for a charity event, a sorority plans to sell a calendar featuring tasteful pictures of some of the more attractive professors on campus. They will need to choose six models from a pool of finalists that includes nine women and six men. How many possible choices are there if they want to feature at least four women?

EXAMPLE 6 Designing a Calendar SOLUTION Since we need to include at least four women, there are three possible compositions: four women and two men, five women and one man, or six women and no men. Four women and two men: Using Fundamental Counting Principle: number of ways to choose 4 women times the number of ways to choose 2 men

EXAMPLE 6 Designing a Calendar SOLUTION Five women and one man: Six women and no men: The total number of possibilities is 1,890 + 756 + 84 = 2,730.

Mrs. Mitchell’s Baseball Team How many ways can 15 out of 31 students be chosen to be on my winning baseball team? Make tally marks to begin counting different groups of 15… How long will this take? We’ll come back to it… Now that my team has been chosen, how many ways can I make a line-up (batting order) of all 15 players? 15! Since only 9 players can be on the field, how many ways can 9 out of 15 players be chosen and put into a batting order? 15P9 At the end of the game, if we didn’t win, then 9 out of 15 players must rake the field. How is this different? (order of the 9 being chosen doesn’t matter) Will the number of groups of 9 rakers be less than, or more than the number of batting orders of 9 out of 15 players? (less than) How do we make the number of combinations less than the number of permutations? Must divide by the number of ways that the 9 chosen to rake the field can be arranged within the same group of 9...9! Connect these concepts to the formulas for nPr and nCr…