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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:

ⁿC_r =	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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