Chapter 10

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# Chapter 10 - PowerPoint PPT Presentation

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

Counting Techniques

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

= 120

10 nCr 3 =

4 nCr 2 x 6 nCr 1 = 6 x 6 = 36

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

121 + 935 + 3635

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:

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