# Why does zero have so much value

## What is Zero-Factorial?

Simple answer: 0! (read "Zero Factorial") is defined to equal 1.

Involved answer(s):

There are several proofs that have been offered to support this common definition.

## Example (1)

If n! is defined as the product of all positive integers from 1 to n, then:

1! = 1*1 = 1

2! = 1*2 = 2

3! = 1*2*3 = 6

4! = 1*2*3*4 = 24

...

n! = 1*2*3*...*(n-2)*(n-1)*n

and so on.

Logically, n! can also be expressed n*(n-1)! .

Therefore, at n=1, using n! = n*(n-1)!

1! = 1*0!

which simplifies to 1 = 0!

## Example (2)

The idea of the factorial (in simple terms) is used to compute the number of permutations (combinations) of arranging a set of n numbers.

n: | Number of Permutations (n!): | Visual example: |
---|---|---|

1 | 1 | {1} |

2 | 2 | {1,2}, {2,1} |

3 | 6 | {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1} |

10 | 3,628,800 | ummm, you get the idea... |

Therefore,

It can be said that an empty set can only be ordered one way, so 0! = 1.

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