Chapter 10 Counting Techniques

Combinations Section 10.3

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

Combination Formula The number of combinations of n objects, taken r at a time(order is not important and n  r).

Combination Formula The number of combinations of n objects, taken r at a time(order is not important n  r).

Combination Rule How many ways can 3 cards be chosen from a standard deck of 52 cards, disregarding the order of the selection? 52 x 51 x 50 3 x 2 x 1 52 nCr 3 = = 22,100

Combination Rule If 20 people all shake hands with each other, how many handshakes are there? 20 x 19 2 20 nCr 2 = = 190 The Greek alphabet has 24 letters. In how many ways can 3 different Greek letters be selected if the order does not matter? 24 x 23 x 22 3 x 2 x 1 24 nCr 3 = = 2024

Combination Rule A committee is to consist of 3 members. If there are 4 men and 6 women available to serve on this committee, find the following: a. How many different committees can be formed? b. How many committees can be formed if each committee must consist of 2 men and 1 woman? 10 x 9 x 8 3 x 2 x 1 10 nCr 3 = = 120 4 nCr 2 x 6 nCr 1 = 6 x 6 = 36

8 nCr 3 + 8 nCr 4 + 8 nCr 5 + 8 nCr 6 + 8 nCr 7 + 8 nCr 8 = Combination Rule How many different committees can be formed from 8 people if each committee must consist of at least 3 people? 8 nCr 3 + 8 nCr 4 + 8 nCr 5 + 8 nCr 6 + 8 nCr 7 + 8 nCr 8 = 56 + 70 + 56 + 28 + 8 + 1 = 219

Combination Rule How many committees of 5 people can be formed from 9 men and 7 women if the committee must consist of less than 3 men? Determine what is acceptable for each gender in order to have a committee of five. Solution: 9 nCr 0  7 nCr 5 + 9 nCr 1  7 nCr 4 +9 nCr 2  7 nCr 3 Acceptable Men Women 121 + 935 + 3635 1 2 5 4 3 21 + 315 + 1260 1596

Combination Rule How many committees of 6 people can be formed from 9 men and 7 women if the committee must consist of more than 4 women? Determine what is acceptable for each gender in order to have a committee of six. Solution: Acceptable Men Women 9 nCr 1  7 nCr 5 + 9 nCr 0  7 nCr 6 921 + 17 1 5 6 189 + 7 Notice 7 is not acceptable for the women. 196 END