Why is 0! equals 1?
Many books will tell you that equals
is a definition. There are actually a few reasons why this is so – the two of which are shown below.
Explanation 1:
Based on my Introduction to Combinations post, we can conclude that taken
at a time is equal to
. This means, that there is only one way that you can group
objects from
objects. For example, we can only form one group consisting of
letters from A, B, C and D using all the 4 letters.
From above, we know that the . But,
. To satisfy the equation,
must be equal to
.
Explanation 2:
We can also use the fact that . Dividing both sides by
, we have
. If we let
, we have
which is what we want to show.
Explanation 3
We can also use the following pattern. We know that which means that
.
Dividing both sides of the equation by , we have
Using this fact, we can check the following pattern.
Now, we go to
As we can see from the 3 examples, .