Explanation on why 0!=1

Yesterday, I posted a tutorial on permutation but I said that I will explain why 0!=1 in my next topic , in case you missed the tutorial click here Permutation tutorial

Now let's start.

From my previous post, I stated that factorial notation formula is;
n! = n * (n - 1) * (n - 2) *..... 3 * 2 * 1

Which will only works when n is greater than 0.

Using the above formula,

6! = 6(6-1)(6-2)(6-3)(6-4)(6-5)

By simplyfying

6!= 6 * 5 *4 *3 * 2 * 1 ---------------eq(1)
Then
6!= 720 ---------------------------------eq(2)

Let's simplify 5!

5!= 5 * 4 * 3 * 2 * 1 ------------------ eq(3)
5!=120
By substituting 5!= 5 * 4 * 3 *2 *1 in eq(1)

Then we will have,

6! = 6 * 5!

By dividing both side by 6 then,

5! = 6!/6

From eq(2) 6!= 720
Substituting that, we the have

5! = 720/ 6
5!= 120
You can see that we get same answer

Do you get it?

Now we can say that,
4! =5!/5

4!= 120/5

4!= 24

Do you agree with that? If you are still confuse, Let's use the above formula to solve it.

4!= 4(4-1)(4-2)(4-3)
4!= 4 * 3 * 2 * 1
4!= 24

Hope you aren't confused anymore?

Now, let's proceed

3!= 4!/4
3!= 24/4
3!= 6

2!= 3!/3
2!= 6/3
2!= 2

1!= 2!/2
1!= 2/2
1!= 1

Then,

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

0!=1

There are some others ways to explain it but it will be more complicated for beginners because it require using advance mathematics topics.

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The picture used in this post was designed by me

References

Numberphile - zero factorial
Math forum - FAQ