1. Exercise
How many different lunches can be made by choosing one of four sandwiches, one of three fruits, and one of two desserts? 24

2. Exercise
How many different lunches can be made by choosing one of four sandwiches, one of three fruits, one of two desserts, and one of two beverages? 48

3. Exercise
How many ways can four separate roles be filled if four people try out? 24

4. Exercise
How many ways can four separate roles be filled if seven people try out?
840

5. Exercise
How many ways can seven separate roles be filled if seven people try out?
5,040

6. Permutation
A permutation is a way of arranging r out of n objects (if r ≤ n).

7. Fundamental Principle of Counting
If there are p ways that a first choice can be made and q ways that a second choice can be made, then there are p × q ways to make the first choice followed by the second choice.

8. n Factorial
The product of n natural numbers from n down to one is called n factorial. The symbol for “factorial” is an exclamation mark, and n! = n(n – 1) … (1). (0! = 1 by definition.)

9. Example 1
Evaluate 6!
6(5)(4)(3)(2)(1)
= 720

10. Example
Evaluate 5!
120

11. Example
Evaluate 8!
40,320

12. Formula for the Permutation of n Objects Taken n at a Time—nPn
To find the number of permutations of n distinct objects taken n at a time, find the product of the positive integers n down through one: nPn = n(n – 1)(n – 2) … (2)(1) = n!

13. Example 2
Find the number of permutations of the letters in the word saved.

14. Formula for the Permutation of n Objects Taken r at a Time—nPr
To find the number of permutations of n distinct objects taken r at a time, use the formula nPr =
n! (n – r)!

15. Example 3
Find the number of permutations of the five letters s, a, v, e, and d taken three at a time.
5! (5 – 3)! =
5 x 4 x 3 x 2 x 1 2 x 1 =
5P3
= 5 x 4 x 3
= 60

16. 8(7)(6)(5)(4)(3)(2)(1) (5)(4)(3)(2)(1)
Example 4
Find the number of permutations of eight distinct things taken three at a time.
8! (8 – 3)! =
8! 5! =
8P3
8(7)(6)(5)(4)(3)(2)(1) (5)(4)(3)(2)(1) =
= (8)(7)(6)
= 336

17. Example
Evaluate 5P3. 60

18. Example
Evaluate 9P4.
3,024

19. Example
Find the number of ways ten books can be arranged on a bookshelf.
10P10 = 10! = 3,628,800

20. Example
Find the number of possible class schedules for a student taking six different classes.
6P6 = 6! = 720

21. Example
Find the number of different ways of choosing 17 committee chairpersons from the 51 senators in the majority party.
51P17 = 51! ÷ 34! = 5.25 × 1027

22. Example
In a race involving six people, how many different orders are possible for the top three finishers?
6P3 = 6! ÷ 3! = 120

23. Example
How many ways are there to select a jury foreman and subforeman from among the twelve jurors?
12P2 = 12! ÷ 10! = 132

24. Example
Find the number of ways of arranging ten books on a bookshelf if five are math books and five are history books and each category must be grouped together.
5! x 5! x 2 = 28,800

25. Exercise
Write an expression in the form nPr to represent the number of permutations in the following situations.

26. Exercise Select first, second, and third place out of 500 contestants.

27. Exercise
Elect a president, vice president, secretary, and treasurer from a class of twenty-eight students.
28P4

28. Exercise
How many different two-digit whole numbers can you make from the digits 2, 4, 6, and 8 if no digit appears more than once in each number?
4P2

29. Exercise
How many different three-letter arrangements are there of the letters of the alphabet?
26P3