n! "n factorial" If nis a positive integer, then n! is nmultiplied by all of the smaller positive integers.
Also,
0! = 1
1! = 1
2! = (2)(1) = 2
3! = (3)(2)(1) = 6
4! = (4)(3)(2)(1) = 24
5! = (5)(4)(3)(2)(1) = 120
6! = (6)(5)(4)(3)(2)(1) = 720
7! = (7)(6)(5)(4)(3)(2)(1) = 5,040
8! = (8)(7)(6)(5)(4)(3)(2)(1) = 40,320
9! = (9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880
10! = (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3,628,800
n! is n multiplied by all of the positive integers smaller than n.
FACT: n! is the number of different ways to arrange (permutations of) n objects.
EXAMPLE 1.5.1 There are four candidates for a job. The members of the search committee will rank the four candidates from strongest to weakest. How many different outcomes are possible?
EXAMPLE 1.5.1 SOLUTION If you were to use the Fundamental Counting Principle, you would need to make four dependent decisions.
1. Choose strongest candidate: 4 options
2. Choose second-strongest candidate: 3 options
3. Choose third-strongest candidate: 2 options
4. Choose weakest candidate: 1 option
(4)(3)(2)(1) = 24
A shorter way to get this answer is to recognize that the problem is asking us to find the number of ways to arrange (according to relative sutability for the job) four people. By definition, the number of ways to arrange 4 things is 4! 4! = 24