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# Why does 0 factorial equal 1?

If you define a factorial operation as repeated multiplications, it makes no sense that 0! = 1 so this causes confusion for the same reason that n0 = 1 also causes confusion. The problem is with the definition. It is restrictive. By using other mathematical ways and logical deduction, we can find out why 0! = 1. Here are a couple of ways:

## Rearranging an equation

It is true to say:
4! = 4 × 3 × 2 × 1

and also:
4! = 4 × (4-1)! because this is 4 × 3! which is 4 × 3 × 2 × 1. Nothing has changed. You can try it with other numbers to confirm:
6! = 6 × (6-1)! = 6 × (5)! = 6 × 5 × 4 × 3 × 2 × 1

So instead of numbers, I use a variable:
n! = n × (n-1)!
Now I divide both sides by n and we get:
 n! = (n-1)! n

Now I can use this to show that 0! = 1. We know that 1! = 1 so if I put the number 1 in place of n we get:
 1! = (1-1)! 1

This is not a proof, just a way of showing one instance where it would make sense that 0! = 1.

## Combinations

There is an equation for calculating the combinations that a number of objects that can be arranged:
 nCr = n! r!(n-r)!
where n is the number of objects in total and r is the number of things you pick out at a time.
Let's look at a number of world leaders in a room. They will need translators but how many?
If there were 4 leaders (n = 4), they would need a translator for 2 (r = 2) people to talk to one another so they would need:
 4! = 4! = 6 translators 2!(4-2)! 2! × 2!

What about 3 leaders?
 3! = 3! = 3 translators 2!(3-2)! 2! × 1!
If you are still unsure, think of an English, Spanish and French leader, they would need an English-French translator, an English-Spanish translator and a Spanish-French translator.

Logically thinking, just two leaders would need only one translator:
 2! = 2! = 1 translator 2!(2-2)! 2! × 0!
so 0! needs to equal 1 to make this equation work

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