EXAMPLE 3.1.20

The Egotists' Club has 6 members: A, B, C, D, E, and F. They are going to line up, from left to right, for a group photo. After lining up in alphabetical order (ABCDEF), Mr. F complains that he is always last whenever they do things alphabetically, so they agree to line up in reverse order (FEDCBA) and take another picture. Then Ms. D complains that she's always stuck next to Mr. C, and that she never gets to be first in line. Finally, in order to avoid bruised egos, they all agree to take pictures for every possible left-to-right line-up of the six people. How many different photos must be taken?

SOLUTION

To arrange the six people in a line we need to make six dependent decisions:

1. Choose first person (6 options);

2. Choose second person (5 options)

3. Choose third person (4 options)

4. Choose fourth person (3 options)

5. Choose fifth person (2 options)

6. Choose last person (1 option)

According to the Fundamental Counting Principle the number of outcomes is

(6)(5)(4)(3)(2)(1) = 720.

Note that the number of ways to arrange 6 people is equal to 6 multiplied all of the positive integers smaller than 6. Likewise it is easy to see that if there were 8 people rather than 6 people, the number of ways to arrange them in a line would be

(8)(7) (6)(5)(4)(3)(2)(1)

and if there were 10 people instead of 6 or 8 the number of ways to arrange them in a line would be (10)(9)(8)(7) (6)(5)(4)(3)(2)(1)

These products are called factorials.

(6)(5)(4)(3)(2)(1) is called “6 factorial” and is denoted 6!

(8)(7) (6)(5)(4)(3)(2)(1) is called “8 factorial” and is denoted 8!

(10)(9)(8)(7) (6)(5)(4)(3)(2)(1) is called “10 factorial” and is denoted 10!

In general, if n is a positive integer, then “n factorial” (denoted n!) is the product of n multiplied by all of the positive integers smaller than n:

n! = (n)(n-1)(n-2)…(3)(2)(1)

Factorials are important in the theory of counting because n! is the number of ways to arrange n objects.