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Chapter 10 Counting Techniques

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Combinations Section 10.3

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**Combinations A selection of distinct objects**

without regard to order is a combination.

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**Combination Formula The number of combinations of n**

objects, taken r at a time(order is not important and n r).

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**Combination Formula The number of combinations of n**

objects, taken r at a time(order is not important n r).

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**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

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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

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**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

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**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 nCr nCr nCr nCr nCr 8 = = 219

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**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 nCr 1 7 nCr 4 +9 nCr 2 7 nCr 3 Acceptable Men Women 1 35 1 2 5 4 3 1596

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**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 nCr 0 7 nCr 6 921 + 17 1 5 6 Notice 7 is not acceptable for the women. 196 END

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Combinations We should use permutation where order matters

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