N factorial

Definition

n factorial, denoted as $n!$, is the product of all positive integers from 1 to n. It is used in permutations, combinations, and other areas involving sequences.

5 Must Know Facts For Your Next Test

1. $0!$ is defined to be 1.
2. Factorials grow very quickly with increasing values of n.
3. $n!$ is used in the formula for permutations: ${}_nP_k = \frac{n!}{(n-k)!}$.
4. $n!$ is also used in the formula for combinations: ${}_nC_k = \frac{n!}{k!(n-k)!}$.
5. The factorial function can be extended to non-integer values using the Gamma function.

Review Questions

• What is the value of $5!$?
• How would you use factorials to calculate the number of ways to arrange 3 out of 7 items?
• Explain why $0!$ equals 1.

Related terms

Permutation: An arrangement of objects in a specific order. Calculated using $${}_nP_k = \frac{n!}{(n-k)!}$$.

Combination: A selection of items without regard to order. Calculated using $${}_nC_k = \frac{n!}{k!(n-k)!}$$.

Gamma Function: An extension of the factorial function that works for non-integer values. Denoted as $\Gamma(n)$, where $(n-1)! = \Gamma(n)$.