What is a factorial

You have 3 seats. Seat A, B and C. You need to find out how many ways you can arrange sitting order for your friends friend 1 through 3 given seats A, B and C. Let's say you choose a seat for a friend at a time. And you follow the order 1 through 3. Friend 1 has 3 seat choices. After you choose one of the 3, 2 are left. So friend 2 has 2 choices. Lastly, friend 3 has 1 choice.

How about we bring it down to two seats A and B. How many ways can these seats be arranged? We have AB and BA. First one can choose from first or second position. Second only has one positin to choose from. That is A_ or B_ then AB or BA.

How about we bring it down to one seat. How many ways can this be arranged? 1 time.

Then zero? How many times can we arrange 0 seats. One time. The arrangement is no arrangement.

So back to three seats. A, B and C.

Take one. We have 3 gaps. We need to fill the first one. Three ways to fill it. A,_,_ or B,_,_ or C,_,_

Take two. We have two gaps. We need to fill the second. Two ways to fill it. [if we chose A for step 1] A,B,_ or A,C,_ [if we chose B for step 1] B,A,_ or B,C,_ [if we chose C step 1] C,A,_ or C,B,_

Take three (one way to feel each gap)

[if we chose AB for slot 1 and 2] A,B,C [if we chose AC for slot 1 and 2] A,C,B [if we chose BA for slot 1 and 2] B,A,C, [if we chose BC for slot 1 and 2] B,C,A [if we chose CA] C,A,B [if we chose CB] C,B,A

So we have 3 ways to fill the first of the 3 empty slots. After which, for each of the three, there's two ways to fill the next slot. And finally we have 1 way to fill the remainder slot for each of the 3x2 That is 3x2x1. That is 3!

A factorial therefore is the number of ways a given number of distinct elements can be arranged.

It is mathematically denoted by the exclamation. A factorial for a number x is x!.