1. Exercise
Evaluate 3! 6

2. Exercise
7! 3!
Evaluate
840

3. Exercise
7! 4!
Evaluate
210

4. Exercise
7! 3!(7 – 3)!
Evaluate 35

5. Exercise
10! 4!(10 – 4)!
Evaluate
210

6. Combination
A combination is a selection of a subset of objects from a set without regard to the order in which they are selected. The notation for the number of combinations of n distinct objects taken r at a time is nCr .

7. permutation The order of the pictures is important.
Example 1
Identify as a permutation or a combination: the number of ways to place five pictures in a line on a wall.
permutation The order of the pictures is important.

8. permutation The office each person fills is important.
Example 1
Identify as a permutation or a combination: the number of ways to fill four eighth-grade class offices from the seven nominees.
permutation The office each person fills is important.

9. combination The committee is the same regardless of order.
Example 1
Identify as a permutation or a combination: the number of ways to choose a five-person class party committee from the twelve volunteers.
combination The committee is the same regardless of order.

10. Example
Identify as a permutation or a combination: the number of ways of choosing two students from a group of forty to be class representatives.
combination

11. Example
Identify as a permutation or a combination: the number of ways of choosing two students from a group of forty to be the class president and vice-president.
permutation

12. Formula for Combinations
To find the number of combinations of n distinct objects taken r at a time, use the formula nCr =
n! r! (n – r)!

13. Example 2
Use the formula to find the number of combinations of three books that Amos could choose from the seven new books that he bought.

14. 7! 3!(7 – 3)! =
7! 3!4! =
7C3
7 × 6 × 5 × 4! 3 × 2 × 1 × 4! =
= 7 × 5
= 35 different combinations

15. Example 3
There are ten girls in Mrs. Hernando’s class, and six are to be selected for a volleyball team. How many different teams can be chosen?

16. 10! 6!(10 – 6)! =
10! 6!4! =
10C6 3 5 2
10 × 9 × 8 × 7 × 6! 6! × 4 × 3 × 2 × 1 =
= 5 × 3 × 2 × 7
= 210 teams

17. Example
Evaluate 7C3. 35

18. Example
Evaluate 12C7.
792

19. Example
Write the answer using combination or permutation notation. Do not evaluate.

20. Example
How many different ways can you choose five out of seven flower types to be included in a bouquet?
7C5

21. Example
How many different ways can a leadoff hitter and a cleanup hitter be chosen from a group of twelve ballplayers?
12P2

22. Example
Find the number of ways of choosing two co-chairs from a list of twelve candidates.
12C2

23. Example
Find the number of ways of selecting a committee of six men and six women from a group of thirty men and twenty-five women.
30C6 × 25C6

24. Example
How many ways can you partition the numbers 1, 2, 3, and 4 into two sets of two numbers each?
4C2 × 2C2

25. Example
How many ways are there to divide a class of eighteen into three equal-size reading groups?
18C6 × 12C6 × 6C6

26. Example
Simplify nC1. n

27. Example
Simplify nCn – 1. n

28. Exercise
The school principal wants to form a committee of five teachers. Twelve of the teachers in the school are women and six are men.

29. Exercise
How many different committees can be formed?
8,568

30. Exercise
How many different all-women committees can be formed?
792

31. Exercise
How many different all-men committees can be formed? 6

32. Exercise
If there were three women on the committee, then two men would have to be chosen to fill the remaining positions on the committee. How many ways can three women be chosen? How many ways can two men be chosen?
220; 15

33. Exercise
Using the Fundamental Principle of Counting, find how many ways a committee of three women and two men can be formed.
3,300

34. Exercise
How many ways can a committee of four women and one man be formed?
2,970