The factorial of a number is the total number of ways in which a set of that many objects can be arranged. See, wasn’t that simple? Let’s try and understand what exactly it’s supposed to mean with the help of mathematics.
First, we’ll have to set some rules (guidelines for how we’re to go about it) because keeping it vague allows a lot of room for misinterpretation. Besides, the beauty of mathematics lies in its precision.
- When we talk of arranging a certain number of objects, the arrangement resembles a cascade or a stack. This means all of the objects are in a line and not in any other unordered formation.For instance, a set of playing cards placed one after another counts as an arrangement. The mess you make, when your house of cards collapses, doesn’t count.
- The arrangement clearly has a start and an end. Every other object, if any, is between those two. The start and the end object in an arrangement will have exactly one neighbouring object while all objects in between will have exactly two neighbouring objects – one before and one after.
- The orientation of the objects themselves don’t matter.You cannot selectively place objects in the same order but different orientations and count that as a different arrangement.
- The distance between two consecutive objects, and thus any two objects, is irrelevant. If placed in the same order, the change in distance doesn’t warrant a count.So, a set of books are only said to be in the same arrangement, if they’re placed in a cascade on one long shelf or multiple shorter shelves but in the same order.
- An arrangement and it’s inverse are not counted as one and the same.Thus, a set of playing cards in ascending order and in descending order are two distinct arrangements for the purposes of this discussion.
- Every arrangement should include every single one of the objects to count as an arrangement.For the arrangement of a set of five objects to count, exactly those five objects have to be used. Using any less or any more will not be a valid count.
In a nutshell, the order alone matters and nothing else.
So, what’s the factorial of one?
For now let’s use the layperson definition of a factorial – the total number of ways a certain number of objects can be arranged. It’s pretty simple. Take one object – a book, a playing card, whatever is in your hand – and place it somewhere in front of you. Observe that it is the start and the end of the arrangement. There is nothing you can do to arrange it in a different way and yet be consistent with the guidelines of this experiment.
Thus it’s concluded that the factorial of one is one. The following is how a mathematician would write it.
1! = 1
The factorial of two
For this we need two objects. Let’s call them object A and object B.
The obvious thing to do is to place A first and B after it. That counts as one arrangement. You can only reverse the order to have A after B.
There you have it. The factorial of two is two.
2! = 2
What about three?
If we have three objects, they can be arranged in the following ways.
Those were all the different arrangements possible for three objects. Turns out there are six ways to do it.
3! = 6
I’d like to, at this point, draw your attention to the first three arrangements. They all have one thing in common: B comes after A. That part is the same as the first arrangement of our experiment to find the factorial of two. Only here we’ve put C in the three places where it could be placed. Let’s illustrate that with a placement table.
|Before A||At A||In between||At B||After B|
As mentioned earlier, the order of the objects are the only thing that matters here. We can do the exact same thing with B before A.
|Before B||At B||In between||At A||After A|
And that’s exactly like the last three arrangements of the table for the factorial of three.
That’s exactly what we did, without noticing it at first, in our experiment to find the factorial of two. We could place B only in the two places around A.