Report

Chapter 10 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 1 Counting Techniques 2 4 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 Combinations Section 10.3 1 2 4 Combinations 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 A selection of distinct objects 1 without regard to order is a combination. 2 4 Combination Formula 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 The number of combinations of n objects, taken r at a time(order is not important and n r). 1 C n r n! ( n r )! r ! 2 4 Combination Formula 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 The number of combinations of n objects, taken r at a time(order is not important n r). 1 C n r P n r r! 2 4 Combination Rule 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 How many ways can 3 cards be chosen from a standard deck of 52 cards, disregarding the order of the selection? 1 2 4 52 nCr 3 = 52 x 51 x 50 = 22,100 3x2x1 Combination Rule 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 If 20 people all shake hands with each other, how many handshakes are there? 20 nCr 2 = 20 x 19 = 190 2 1 2 The Greek alphabet has 24 letters. In how many ways can 3 different Greek letters be selected if the order does not matter? 24 nCr 3 = 24 x 23 x 22 = 2024 3x2x1 4 Combination Rule 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 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: 1 2 a. How many different committees can be formed? 10 x 9 x 8 = 120 10 nCr 3 = 3x2x1 4 b. How many committees can be formed if each committee must consist of 2 men and 1 woman? 4 nCr 2 x 6 nCr 1 = 6 x 6 = 36 Combination Rule 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 How many different committees can be formed from 8 people if each committee must consist of at least 3 people? 1 2 8 nCr 3 + 8 nCr 4 + 8 nCr 5 + 8 nCr 6 + 8 nCr 7 + 8 nCr 8 = 4 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? 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 Determine what is acceptable for each gender in order to have a committee of five. Acceptable Men Women 0 5 1 4 2 3 1 Solution: 2 9 nCr 0 7 nCr 5 + 9 nCr 1 7 nCr 4 +9 nCr 2 7 nCr 3 4 121 + 935 + 3635 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? 00 11 0 010 1 01 0 110 1 00 01 01 00 1 011 Determine what is acceptable for each gender in order to have a committee of six. Acceptable Men Women 1 5 0 6 Notice 7 is not acceptable for the women. Solution: 1 2 4 9 nCr 1 7 nCr 5 + 9 nCr 0 7 nCr 6 921 + 17 189 + 7 196 END