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**,

**and**

*B, C***.**

*D*using all the 4 lettersFrom 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, .