## Why 0!=1.

Why 0!=1, or we can say, “Wow! 0! = 1    !!!!!,” or we can say, “Wow,!, 0!=1!”  But why?

From Mathandmultimedia.com

Many books will tell you that $0!$ equals $1$ is a definition. There are actually a few reasons why this is so – the two of which are shown below.

Explanation 1:

Based on my Introduction to Combinations post, we can conclude that $n$taken $n$ at a time is equal to $1$. This means, that there is only one way that you can group $n$ objects from $n$ objects. For example, we can only form one group consisting of $4$ letters from AB, C and using all the 4 letters.

From above, we know that the $\displaystyle{n \choose n} = 1$. But, $\displaystyle{n \choose n} = \displaystyle\frac{n!}{(n-n)!n!} = \displaystyle\frac{n!}{0!n!} = 1$. To satisfy the equation, $0!$ must be equal to $1$.

Explanation 2:

We can also use the fact that $n! = n(n-1)!$. Dividing both sides by $n$, we have $\displaystyle\frac{n!}{n} = (n-1)!$. If we let $n = 1$, we have $\displaystyle\frac{1!}{1} = (1 - 1)! \Rightarrow 1 = 0!$ which is what we want to show.