Why the factorial of 0 is 1

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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.

Factorial guidelines

  • 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.

ArrangementsCount
1st2nd
AB1
BA2

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.

ArrangementsCount
1st2nd3rd
ABC1
ACB2
CAB3
BAC4
BCA5
CBA6

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.

Placement Table
Before AAt AIn betweenAt BAfter B
-A-BC
-ACB-
CA-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.

Placement Table
Before BAt BIn betweenAt AAfter A
-B-AC
-BCA-
CB-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.

Simplification of factorial finding

Using this new information it is easy to find the factorial of larger numbers. Here are a few things to remember.

  • A set of objects that are placed in a certain order or arrangement will have more gaps than the number of objects by one.One object has two gaps that could allow an additional object. Two objects will have three gaps that can be filled by a third object. The same goes for large numbers. Twenty objects will have twenty one different places where a new object can be added.
  • Keeping that in mind, it appears if we already know the factorial of N objects (say N!), the factorial of N+1 objects is simply the product of N+1 and the factorial of N.

    (N+1)! = (N+1)•N!

That formula seems to work correctly with finding the factorial of three when the factorial of two is known. Since,

2! = 2

it should follow that

3! = 3(2!) = 3(2) = 6

which is true as we’ve discovered just now.

We can calculate the factorials of 4 to 10 similarly.

4! = 4(3!) = 4(6) = 24

5! = 5(4!) = 5(24) = 120

6! = 6(5!) = 6(120) = 720

7! = 7(6!) = 7(720) = 5040

8! = 8(7!) = 8(5040) = 40320

9! = 9(8!) = 9(40320) = 362880

10! = 10(9!) = 10(362880) = 3628800

Needless to say, factorials get too large too soon.

Factorial of four

Let’s see if the simplified method for finding factorials works for four, which by the looks of it seems only 20% as hard as the factorial of five.

ArrangementCount ArrangementCount
1st2nd3rd4th1st2nd3rd4th
ABCD1 BACD13
ABDC2BADC14
ADBC3BDAC15
DABC4DBAC16
ACBD5BCAD17
ACDB6BCDA18
ADCB7BDCA19
DACB8DBCA20
CABD9CBAD21
CADB10CBDA22
CDAB11CDBA23
DCAB12DCBA24

Those were the twenty four unique ways one could arrange four objects.

Factorial of five

To arrange five objects one would have to place four objects in any one of those orders and the fifth in any of the five available places. Doing that for every single order would yield a total of 120 ways in which five objects could be arranged. Let’s use a simpler table here.

Base Arrangements
1st2nd3rd4th5th
ABCD ABCDEABCEDABECDAEBCDEABCD
ABDCABDCEABDECABEDCAEBDCEABDC
ADBCADBCEADBECADEBCAEDBCEADBC
DABCDABCEDABECDAEBCDEABCEDABC
ACBDACBDEACBEDACEBDAECBDEACBD
ACDBACDBEACDEBACEDBAECDBEACDB
ADCBADCBEADCEBADECBAEDCBEADCB
DACBDACBEDACEBDAECBDEACBEDACB
CABDCABDECABEDCAEBDCEABDECABD
CADBCADBECADEBCAEDBCEADBECADB
CDABCDABECDAEBCDEABCEDABECDAB
DCABDCABEDCAEBDCEABDECABEDCAB
BACDBACDEBACEDBAECDBEACDEBACD
BADCBADCEBADECBAEDCBEADCEBADC
BDACBDACEBDAECBDEACBEDACEBDAC
DBACDBACEDBAECDBEACDEBACEDBAC
BCADBCADEBCAEDBCEADBECADEBCAD
BCDABCDAEBCDEABCEDABECDAEBCDA
BDCABDCAEBDCEABDECABEDCAEBDCA
DBCADBCAEDBCEADBECADEBCAEDBCA
CBADCBADECBAEDCBEADCEBADECBAD
CBDACBDAECBDEACBEDACEBDAECBDA
CDBACDBAECDBEACDEBACEDBAECDBA
DCBADCBAEDCBEADCEBADECBAEDCBA

This table clearly shows 120 entries to the number of arrangements, which should be obvious if you know your tables of 5 or that of 24.

From all the information gathered, let’s start to define a factorial as it’s done with mathematics.

Factorial defined

We’ve seen that the factorial of N+1 is the product of the factorial of N and the number N+1.

(N+1)! = (N+1)•N!

Similarly one could also write the following:

N! = N•(N-1)!

That could be further written as

N! = N•(N-1)•(N-2)!

And this could continue to

N! = N•(N-1)•(N-2)•…•(3)•(2)•(1!)

The factorial of a number can thus be defined as the product of every natural number upto and including itself.

Now, here’s the thing. Once you arrived at an expression that stops at 1!, you can either just take that as equal to 1, or go one step further and substitute that with the product of 1 and 0!.

N! = N•(N-1)•(N-2)•…•(3)•(2)•(1)•(0!)

We can thus deduce what exactly the factorial of zero from this expression.

1! = 1•0!

Since the factorial of one is one and one times any number is that number we get

0! = 1

Thus, after a long and arduous process, we’ve proved that the factorial of zero is indeed one using mathematics.

Can it be made simpler?

Of course, it can. Knowledge of basic maths wasn’t really required here. As stated in the very first line, the factorial of a number is the total number of ways in which a set of that many objects can be arranged. One needn’t be a math enthusiast to know that there is only one way to arrange zero objects.

Thus, the factorial of zero is one.

Do let me know via the comments what you thought about this article. You can make suggestions for topics you think I should discuss. Thanks for reading.