Exercise Evaluate 3! 6

Exercise 7! 3! Evaluate 840

Exercise 7! 4! Evaluate 210

Exercise 7! 3!(7 – 3)! Evaluate 35

Exercise 10! 4!(10 – 4)! Evaluate 210

Combination A combination is a selection of a subset of objects from a set without regard to the order in which they are selected. The notation for the number of combinations of n distinct objects taken r at a time is nCr .

permutation The order of the pictures is important. Example 1 Identify as a permutation or a combination: the number of ways to place five pictures in a line on a wall. permutation The order of the pictures is important.

permutation The office each person fills is important. Example 1 Identify as a permutation or a combination: the number of ways to fill four eighth-grade class offices from the seven nominees. permutation The office each person fills is important.

combination The committee is the same regardless of order. Example 1 Identify as a permutation or a combination: the number of ways to choose a five-person class party committee from the twelve volunteers. combination The committee is the same regardless of order.

Example Identify as a permutation or a combination: the number of ways of choosing two students from a group of forty to be class representatives. combination

Example Identify as a permutation or a combination: the number of ways of choosing two students from a group of forty to be the class president and vice-president. permutation

Formula for Combinations To find the number of combinations of n distinct objects taken r at a time, use the formula nCr = . n! r! (n – r)!

Example 2 Use the formula to find the number of combinations of three books that Amos could choose from the seven new books that he bought.

7! 3!(7 – 3)! = 7! 3!4! = 7C3 7 × 6 × 5 × 4! 3 × 2 × 1 × 4! = = 7 × 5 = 35 different combinations

Example 3 There are ten girls in Mrs. Hernando’s class, and six are to be selected for a volleyball team. How many different teams can be chosen?

10! 6!(10 – 6)! = 10! 6!4! = 10C6 3 5 2 10 × 9 × 8 × 7 × 6! 6! × 4 × 3 × 2 × 1 = = 5 × 3 × 2 × 7 = 210 teams

Example Evaluate 7C3. 35

Example Evaluate 12C7. 792

Example Write the answer using combination or permutation notation. Do not evaluate.

Example How many different ways can you choose five out of seven flower types to be included in a bouquet? 7C5

Example How many different ways can a leadoff hitter and a cleanup hitter be chosen from a group of twelve ballplayers? 12P2

Example Find the number of ways of choosing two co-chairs from a list of twelve candidates. 12C2

Example Find the number of ways of selecting a committee of six men and six women from a group of thirty men and twenty-five women. 30C6 × 25C6

Example How many ways can you partition the numbers 1, 2, 3, and 4 into two sets of two numbers each? 4C2 × 2C2

Example How many ways are there to divide a class of eighteen into three equal-size reading groups? 18C6 × 12C6 × 6C6

Example Simplify nC1. n

Example Simplify nCn – 1. n

Exercise The school principal wants to form a committee of five teachers. Twelve of the teachers in the school are women and six are men.

Exercise How many different committees can be formed? 8,568

Exercise How many different all-women committees can be formed? 792

Exercise How many different all-men committees can be formed? 6

Exercise If there were three women on the committee, then two men would have to be chosen to fill the remaining positions on the committee. How many ways can three women be chosen? How many ways can two men be chosen? 220; 15

Exercise Using the Fundamental Principle of Counting, find how many ways a committee of three women and two men can be formed. 3,300

Exercise How many ways can a committee of four women and one man be formed? 2,970