factorial() — Factorial

Counts the different ways you can line x things up in order.

factorial(x)

Key idea

• A factorial x! means: multiply x by every whole number below it, all the way down to 1. (e.g. 5! = 5x4x3x2x1 = 120)
• It counts the ways to put x different items in order in a row. (e.g. 3 books -> 3! = 6 orders)
• By definition 0! equals 1, since there is exactly one way to arrange nothing. (e.g. 0! = 1)
• Factorials grow extremely fast as x gets bigger. (e.g. 10! = 3,628,800)
• Each step just multiplies the previous result by the next number. (e.g. 4! = 3! x 4 = 24)

Definition

$$x!=\prod _{i=1}^{x}i=x(x-1)\cdots 2\cdot 1$$

$x$

how many distinct items

$x!$

x times (x-1) times ... times 1

$$5!=5\cdot 4\cdot 3\cdot 2\cdot 1=120$$

Where you'd use it

• You want to line up 5 friends for a photo and ask how many different orders are possible. factorial(5) = 5 × 4 × 3 × 2 × 1 = 120, so there are 120 distinct line-ups.
• A DJ has 6 songs to play and wonders how many running orders exist. factorial(6) = 720 different set lists.
• To seat 4 guests around a row of chairs, the number of arrangements is factorial(4) = 24, showing how fast the count climbs as guests are added.

Frequently asked questions

What is a factorial used for?

Counting arrangements (n!). It underlies permutations and combinations.

Why did I get Inf?

Above about 170, n! overflows R’s number range.