# What is the factorial of 0?

Aug 14, 2014

It is 1 by definition. However, this definition fits in nicely with applications for factorials.

Since you have posted this under binomial theorem, let's look at this application. For example: ${\left(x + 1\right)}^{3}$.

Expanding this, we get: $1 {x}^{3} + 3 {x}^{2} + 3 x + 1$

The coefficients are determined by $1 {=}_{3} {C}_{0}$, $3 {=}_{3} {C}_{1}$, $3 {=}_{3} {C}_{2}$, $1 {=}_{3} {C}_{3}$.

${.}_{n} {C}_{r}$ is $n$ choose $r$ and defined as (n!)/(n!(n-r)!.

So, ._3 C_0=(3!)/(3!(3-0)!)=(3!)/(3!0!). If 0! =0, then ${.}_{3} {C}_{0}$ would be undefined. However, when we use 0! =1, ${.}_{3} {C}_{0} = 1$ which gives us the proper coefficient. This explains why we want to define 0! =1.