Why does 0!=1?

Question

I dont understand why 0! (factorial) =1.
How can 0x0 =1??

Answer 1

Usually n factorial is defined in the following way:

n! = 1*2*3*...*n
But this definition does not give a value for 0 factorial, so a natural question is: what is the value here of 0! ?

A first way to see that 0! = 1 is by working backward. We know that:

1! = 1
2! = 1!*2
2! = 2
3! = 2!*3
3! = 6
4! = 3!*4
4! = 24


We can turn this around:
4! = 24
3! = 4!/4
3! = 6
2! = 3!/3
2! = 2
1! = 2!/2
1! = 1
0! = 1!/1
0! = 1


In this way a reasonable value for 0! can be found.
How can we fit 0! = 1 into a definition for n! ? Let's rewrite the usual definition with recurrence:

1! = 1
n! = n*(n-1)! for n > 1


Now it is simple to change the definition to include 0! :
0! = 1
n! = n*(n-1)! for n > 0


Why is it important to compute 0! ?
An important application of factorials is the computation of number combinations:

n!
C(n,k) = --------
k!(n-k)!


C(n,k) is the number of combinations you can make of k objects out of a given set of n objects. We see that C(n,0) and C(n,n) should be equal to 1, but they require that 0! be used.
n!
C(n,0) = C(n,n) = ----
n!0!


So 0! = 1 neatly fits what we expect C(n,0) and C(n,n) to be.

Answer 2

X! = x*(x-1)*(x-2) and so on with the condition that none of the terms will be zero. 0! is a special case and simply states that the product of no numbers at all is 1. It is a condition, not a calcuation. As to bluesky's proof, you cannot divide both sides by (a-b) because a-b=0 since a = b.

Answer 3

n! is the number of ways to arrange n objects in order. 0! is 1 because there is only one way to arrange 0 objects. But it really doesn't have anything to do with 0*0.

Technically n! is defined this way:

n! =
1, n = 0
n*(n-1)!, n = 1, 2, 3, ...

So 0! = 1
1! = 1*(1-1)! = 1*0! = 1*1 = 1
2! = 2*(2-1)! = 2*1! = 2*1 = 2
3! = 3*(3-1)! = 3*1! = 3*2 = 6

and so on.

Answer 4

Actually, 0! = 1 by definition. But like others have pointed out, it's a very interesting definition, because it makes many formula work when n =0.

Answer 5

Experimenting with factorials, we come up with n!=n(n-1)!. For example 17!=17x(16!):

16!=1x2x...x16
17!=(1x2x...x16)x17

That equation (n!=n(n-1)!) just dictated to us where to put the parentheses. By making n=1, we can find 0!:

1!=1(0!)
0!=1

And, it turns out that 0!=1 works very well in many situations (in probability, for example).

Answer 6

In the 'C' programming language '!' means NOT and = means equal.

1 NOT Equal 0
So 1 != 0.

Answer 7

The same reason 0^0=1

Answer 8

Because there is only 1 order in which you can arrange 0 items.

Cheers.

Answer 9

Are you sure you are not mistaking for the programming paradigm 0 is not equal to 1 (with != meaning not equal to)

Answer 10

I just did this in class today..but that still doesn't mean that I know what I'm talking about. I think that in actuality is the first number, so that's why...I might be wrong, but it's just math, it won't kill you to be wrong.