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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

Chapter 10

Counting Techniques

slide3

Combinations

  • A selection of distinct objects
  • without regard to order is a
  • combination.
slide4

Combination Formula

  • The number of combinations of n
  • objects, taken r at a time(order is
  • not important and nr).
slide5

Combination Formula

  • The number of combinations of n
  • objects, taken r at a time(order is
  • not important nr).
slide6

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

slide7

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

slide8

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

slide9

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

slide10

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

slide11

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