Why is the Factorial of Zero Defined as One?

[ananthu017]

can u give me the method to find the factorial of zero ?

 

[Borek]

What is the definition of the factorial?

 

[DeIdeal]

$0!=\Gamma(1)=\intop_{0}^{\infty} t^{1-1}\exp(-t)\mathrm{d}t=-[\exp(-t)]_{0}^{\infty}=-(0-1)=1.$

Alternatively, there's exactly one way to arrange an empty set.

 

[Mentallic]

$0!=\Gamma(1)=\intop_{0}^{\infty} t^{1-1}\exp(-t)\mathrm{d}t=-[\exp(-t)]_{0}^{\infty}=-(0-1)=1.$

I highly doubt the OP would be able to make heads or tails of this.

$$n! = n(n-1)!$$

Work with this definition, it's all you need.

 

[HallsofIvy]

So your answer to the original question is that $0!= 0(-1)!$? But what is (-1)!?

(Yes, you can write that 1!= 1(0!) and, if you know that 1!= 1, then it follows that 0!= 1. But better is just to state the basic definition of n! that asserts 0!= 1 for as part of the definition.)