Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A factorial, denoted by $n!$, is the product of all positive integers less than or equal to $n$. It is used in permutations, combinations, and other mathematical calculations.
5 Must Know Facts For Your Next Test
The factorial of zero ($0!$) is defined as 1.
Factorials grow very quickly; for example, $5! = 120$ and $10! = 3,628,800$.
$n!$ can be computed recursively: $n! = n \times (n-1)!$ with the base case of $0! = 1$.
Factorials are used in the binomial theorem for calculating coefficients in binomial expansions.
In sequences and series, factorials are often used to describe terms in exponential or polynomial expressions.
An arrangement of a set of objects in a specific order. The number of permutations of $n$ distinct objects taken $r$ at a time is given by $_{n}P_{r} = \frac{n!}{(n-r)!}$.
A selection of items from a larger pool where order does not matter. The number of combinations of $n$ items taken $r$ at a time is given by $_{n}C_{r} = \frac{n!}{r!(n-r)!}$.
A coefficient that appears in the binomial theorem, represented as $\binom{n}{k}$ or $_{n}C_{k}$ and calculated using $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.