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.
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 .
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.
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, .