How To Arrange Nothing?

By the way, what even is arrangement?

This is a tricky but an interesting question. Before we answer it, one might as well ask what it means to ‘arrange’ anything in the first place. How does one mathematically formulate the arrangement of things? It boils down to an operator in mathematics called a factorial.

For those who do not know, a factorial is a unary operator, that anyone with basic multiplication skills (or a calculator) can perform. In combinatorics, performing this operation on a finite set gives you the number of possible ways one can arrange the given set. Simply put, it tells you in how many ways you can arrange things. For example, let us pretend that I have 3 cards of distinct color (say red, green and blue). How many ways can I arrange these cards? This is quite simple. Since I already have the cards with me (which was the assumption I made before I asked the question), I can organize them into a batch of ordered tuples whose order is unique to each other. To put it to perspective, I could arrange them in the following ways:

Figure 1.1: Possible arrangements of 3 cards.

Since there are only 3 cards, this result is empirically verifiable with not just cards, but 3 distinct objects of any sort. As observed in Figure 1.1, there are 6 unique ways to arrange those cards. But as you will see later on, this method of counting every unique instance of arrangement of objects is painful and very inefficient. Let us say, I had 5 cards instead of 3. Now the number of possible unique arrangements is nowhere near 6 (Fancy a challenge? Try it out yourself!). Needless to say, the process of physically arranging objects become tedious for numbers like 5 and its successors. So, what I propose is a different method (which becomes the only practical method as the number of elements in the set go up). For comparison, let us go back to the initial example of 3 unique cards. Factorial of 3 can be written as:

Figure 1.2: Factorial of 3

The exclamation (!) mark represents the factorial. As you can see, the factorial of any number is the product of itself and its preceding natural numbers till 1. So, the factorial of 3 becomes:

Figure 1.3: Factorial of 3 (final result)

This exposes an interesting result. The number of ways one could arrange 3 unique cards is equivalent to the factorial of the number of cards, which in this case was 3. Needless to say, this is not a specific case for cards; any objects will do. In fact, it does not even have to be an object. In set theory, we have what we call elements. A set contains n number of elements, similar to how I can hold n number of cards in my hand. That means that we no longer have to arrange the elements by ourselves; instead, we have the factorial to help us out. In general, for any n number of elements:

Equation 1.1: Factorial of n

Let us call this equation 1.1. If I reversed the order (since multiplication is commutative), it becomes:

Equation 1.2: Factorial of n, in reverse order

To test this out, let us go back to the case of 5 cards. Taking the factorial of 5, one would get to:

Figure 1.4: Factorial of 5

Hence, there are 120 ways I can uniquely arrange 5 cards. Clearly, attempting to compute them by hand is an impractical approach.

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Now back to our question of interest, how do we arrange nothing? Till this moment, we had something to arrange, and now that we know factorials, we also possess a method to compute the number of arrangements of something. But what is nothing? There is only one number in the entire number system we can think of that has the duality of existing as a numerical value and as the representation of absence of one, which is obviously zero. But this poses a problem; a factorial can only figure out the number of permutations, as long as we deal with natural numbers. But zero is a whole number. In essence, our question boils down to finding the factorial of 0, represented by 0!.

There are two ways to approach this problem. While in a conversation, one who is interested in combinatorics might argue that the only possible way one can arrange nothing would be only one way, which is by doing, well, nothing. This does not hold any mathematical rigor, yet it grips the essence of the result which we will get to in a moment. But right now, think about what I just said. The minimum number of permutation something, let alone nothing, can achieve is one. Why is it not zero? Factually, doing nothing is also considered a permutation. But if that was the case, why is it that 3! = 6 and not 7? One way to think about it would be that if we do nothing and call it an arrangement by itself in the case of the 3 cards, we also make the implicit statement that the number of cards we ought to arrange, which was 3, is now zero — because technically, we have not arranged anything. The same goes for nothing as well, except in this case, that statement is right (because for an empty set , there exists precisely one arrangement — the empty tuple itself!).

The above statement is just a conceptual result of what we ought to prove right now. To find the factorial of 0, we ought to begin by rewriting the general equation of a factorial:

Equation 1.3: General formula for factorial of n, reinterpreted.

Note that this is just another way of writing the equation 1.1 which I mentioned earlier. Now, I can rearrange this equation in any way I want:

Now it is just a matter of logic. Instead of plugging 0 to n directly, one might as well plug in 1. This naturally generates 0! on the left, and the whole equation becomes:

This holds true because factorial of 1 is 1 itself, since there is not more than one way of arranging a single element. This gives us our desired result.

If you are familiar with set notation, you can easily derive this result in yet another way by considering an empty set (null set). There is only a single possible way to arrange elements inside a null set, since there are no elements to arrange. But come on, now you know what factorials are!

Now that you know what a factorial is, here is a list of things you can go ahead and look up right away:

· Till this point, we have been under the assumption that factorials exist only for natural numbers. How do you find the factorial of a decimal, a negative integer, or any rational or irrational number for that matter? (Look up for Gamma function.)

· Factorials have a wide range of applications in daily life. For a heads-up, search for how websites determine the security of your passcode, or how sports or e-sports tournaments make use of this operator for matchups.