Answer:
No, I'm not particularly excited about this answer. That exclamation point, when used in mathematics, is called a factorial. It tells you to take the number in front of it, in this case 5, and multiply it by every digit smaller than it until you get to 1. So 5! = 5x4x3x2x1 = 120.
How did I get that answer? Well, imagine that you have your five empty chairs in front of you waiting to do what they do best: seat people.
\__/ \__/ \__/ \__/ \__/
Now, let's say there are 5 people named Alice, Bob, Charlie, Damon, and Eliza (who all like to go by their first initial), and they all want to have a seat. Well, how can you arrange them?
To start, we know that someone can only sit in one seat at a time. We can also imagine that the seats are only big enough for one person to sit in.
Let's say B sits in seat one.
\B/ \__/ \__/ \__/ \__/
Who can we put in the seat next to B? Either A, C, D, or E. Let's say we know that E and B are good friends, so we sit E in seat two.
\B/ \E/ \__/ \__/ \__/
Who can we place next to E? Well, we have three options left: A, C, and D. Perhaps A has a crush on E, so A can sit in seat three.
\B/ \E/ \A/ \__/ \__/
Now for seat number four, we can fill it with either C or D. Let's say D is impatient with our slow process to fill the seats, so they sit down.
\B/ \E/ \A/ \D/ \__/
Finally, we can sit poor, tired C down in seat number 5.
\B/ \E/ \A/ \D/ \C/
You might be thinking to yourself, "Great, he's only shown me one way to assign 5 people to 5 chairs! What if A goes in seat one, B in two, C in three, D in four, and E in five?"
Well, if that's what you're thinking, then you're right! In fact, we can pick any of the five to go in seat one, then whoever's left of the four people in seat two, any of the three people for seat three, and so on.
Aha! 5…4…3…2…1. This looks just like 5! without those multiplication signs gluing everything together. Now the question becomes: how do we know to multiply the choices together, rather than add them?
The answer comes from what's known as the Fundamental Counting Principle (FCP). It says that when you have N different ways to do one thing and M ways to do something else, then the total number of ways you can do them all together is NxM. So for your problem, you have 5 ways to fill seat one, 4 ways to fill seat two, 3 ways to fill seat three, 2 ways to fill seat four, and 1 way to fill seat five. 5x4x3x2x1 different ways by the FCP.
