Prove that 0! = 1
It is defined that way. It makes possible the definition that
n! = n(n-1)! for all integer n>0
Suppose we have n objects.
In how many ways can I take all of them
of course the answer to that is 1 way.
but from the theory of combinations it would be
n!/( (n-n)(n!) )
= n!/(0!n!)
= 1/0!
but we know that has to be 1
1/0! = 1, this can only be true if we define 0! = 1
That is , we are forced to accept 0! = 1 if our theory of combinations and permutations is accepted.
Thanks for the help !
To prove that 0! (zero factorial) is equal to 1, we can use the definition of factorial.
Factorial is defined for non-negative integers n as the product of all positive integers less than or equal to n. For example, 4! (read as "4 factorial") is calculated as 4 x 3 x 2 x 1 = 24.
By this definition, we can see that 0! should be the product of all positive integers less than or equal to 0. However, there are no positive integers less than 0, so if we follow the definition literally, it would suggest that 0! is an empty product, which is undefined.
However, there is another way to reason about 0!. We can consider it as a special case or base case.
First, let's consider the pattern we observe when calculating factorials:
1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
We notice that as we calculate the factorial for each successive positive integer, we multiply it by the previous factorial. So we could write n! = n x (n-1)!
Using this pattern, we can calculate 0! as follows:
0! = 1 x (0-1)! = 1 x (-1)!
We encounter a problem here because factorial is not defined for negative numbers. However, this is where the special case of 0! comes into play.
Since there are no positive integers less than 0, we can consider 0! as a base case where we stop the "cascading" multiplication at 1. This means that by convention, we define 0! to be equal to 1.
So even though there is no direct mathematical proof using the definition of factorial, we can reason and justify that 0! is defined as 1 based on the pattern and special case considerations.
Therefore, we can conclude that 0! = 1.