Question

What is the factorial of 0?

Answer 1

The simple answer is 1.

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: `(x+1)^3`.

Expanding this, we get: `1x^3+3x^2+3x+1`

The coefficients are determined by `1=_3 C_0`, `3=_3C_1`, `3=_3C_2`, `1=_3C_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`.