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Chapter 10

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 ). .

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Chapter 10

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  1. Chapter 10 Counting Techniques

  2. CombinationsSection 10.3

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

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

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

  6. 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

  7. 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

  8. 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 = 120 10 nCr 3 = 4 nCr 2 x 6 nCr 1 = 6 x 6 = 36

  9. 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

  10. 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 121 + 935 + 3635 21 + 315 + 1260 1596

  11. 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: 9 nCr 1  7 nCr 5 + 9 nCr 0  7 nCr 6 921 + 17 189 + 7 Notice 7 is not acceptable for the women. 196 END

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