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, .

But this means that 1! = 0!, isn’t it? Weird…

Yes, it’s weird. 🙂

Thank you so much for this first example using the formula for combinations. I have had a lot of trouble explaining this to some of the home schooled students that I tutor from time to time, and I’ve never found an explanation before now that really set right with me for showing advanced high school-aged children.

I like it!

Thanks Ceesay. 😀

Awesome! Thanks sir..

0!=1

prove

n!=n(n-1)!

n!/(n-1)!=n

5!/4!=5

4!/3!=4

3!/2!=3

2!/1!=2

1!/0!=1

0!=1!/1=1/1=1

gamma fn of ( m+1) is m!

gamma fn of (1) is 0! (By definition of gamma fn)

0!=1 (gamma fn of 1 is 1)

Accurate proof