1. Chapter 10
Counting Techniques

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

9. 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

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 nCr 1  7 nCr 4 +9 nCr 2  7 nCr 3
Acceptable
Men
Women
1  35 1 2 5 4 3
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:
Acceptable
Men
Women
9 nCr 1  7 nCr nCr 0  7 nCr 6
921 + 17 1 5 6
Notice 7 is not acceptable for the women.
196
END