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.
