What is the factorial of 0?

1 Answer
Amory W.
Aug 14, 2014

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.

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