Mathematics and Statistics · Ch 15 — Permutations and Combinations

Factorial Notation

Counting arrangements produces products such as $5 \times 4 \times 3 \times 2 \times 1$ so often that a short symbol is used for them.

Note

Factorial

For a positive integer $n$ , the symbol $n!$ (read " $n$ factorial") means the product of all positive integers from $1$ up to $n$ :

$n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1.$

By convention $0! = 1$ (this keeps the permutation and combination formulas below consistent).

For example $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ , and $3! = 6$ , $4! = 24$ , $6! = 720$ .

Note

The Key Recursive Property

$n! = n \times (n-1)!$

Because $n!$ is just $n$ times the product that makes up $(n-1)!$ . This lets a large factorial be written in terms of a smaller one, which is how factorial expressions are simplified without multiplying everything out. …

Definition 3Factorial ($n!$)

$n! = n \times (n-1) \times \cdots \times 2 \times 1$ for a positive integer $n$ , with $0! = …

Definition 4Recursive property

$n! = n \times (n-1)!$ , which allows a larger factorial to be expressed through a smaller one and factorial fractions to be si …

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