Factorial notation

The definition of n!, why 0! = 1, unrolling factorials, and simplifying and cancelling expressions with factorials.

The definition

$n$	0	1	2	3	4	5	6	7	8
$n!$	1	1	2	6	24	120	720	5040	40 320

Factorials grow extremely fast: $10! = 3\,628\,800$ and $20!$ has $19$ digits. That is why we cancel them rather than evaluate them.

Example 1●

Evaluate $\dfrac{10!}{7!}$.

$\dfrac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 = 720$

Example 2●●

Simplify $\dfrac{(n + 1)!}{(n - 1)!}$.

$\dfrac{(n + 1)\,n\,(n - 1)!}{(n - 1)!} = n(n + 1)$

Example 3●●●

Write $\dfrac{1}{5!} + \dfrac{1}{6!}$ as a single fraction, and simplify $\dfrac{n!}{(n - 2)!} = 42$ to solve for $n$.

$\dfrac{6}{6!} + \dfrac{1}{6!} = \dfrac{7}{720}$

Common denominator 6! = 6 × 5!.

$\dfrac{n!}{(n - 2)!} = n(n - 1) = 42$, so $n^2 - n - 42 = 0$, $(n - 7)(n + 6) = 0$, and $n = 7$.

Flashcards

$n!$

$n(n - 1)(n - 2)\cdots 2 \times 1$

$0!$

$1$

$dfrac{n!}{(n - 1)!}$

$n$

$dfrac{n!}{(n - 2)!}$

$n(n - 1)$

Practice