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Why Does Zero Factorial Equal One?

Find Out Why 0! = 1

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The factorial is a short hand way to write the multiplication of successive decreasing natural numbers. For example, rather than write 4 x 3 x 2 x 1, we use an exclamation mark to denote this quantity as 4!. It is pretty clear from this definition how to calculate the factorial for any number greater than or equal to one; we can see that 2! = 2 x 1 = 2, and 1! =1.

But what is the value of zero factorial? After all, zero is a natural number so we should be able to find the factorial for this number. Also in the definition of factorial is that 0! = 1. This typically makes people a little confused the first time that they see this. Since zero times anything is zero, shouldn’t zero factorial be zero? There are a few reasons why 0! = 1 makes sense.

By Definition

The first reason for why zero factorial is equal to one is because this is what it the definition says it should be. This is ultimately the only reason why, and is a mathematically correct explanation. However, this answer obscures some things behind the scenes. If we can define zero factorial to be one, then why not define 0! = 0 to keep it consistent with the definition of factorial for the other natural numbers? It turns out that when we look at some counting arguments, the definition of 0! = 1 makes perfect sense.

Permutations

A permutation is where we order elements in a set. For example there are six permutations of the set {1, 2, 3} since we may write these three elements in the following six ways:

  • 1, 2, 3
  • 1, 3, 2
  • 2, 3, 1
  • 2, 1, 3
  • 3, 2, 1
  • 3, 1, 2
There are three elements in the set that we are looking at, and six permutations of this set. We could also state this fact through the equation 3! = 6.

In a similar way there are 4! = 24 permutations of a set with four elements and 5! = 120 permutations of a set with five elements. So an alternate way to think about the factorial is to let n be a natural number and say that n! is the number of permutations for a set with n elements.

With this way of thinking about the factorial, let’s look at a couple more examples. A set with two elements has two permutations: {a, b} can be arranged as a,b or as b,a. This corresponds to 2! = 2. A set with one element has a single permutation, as the element 1 in the set {1} can only be ordered in one way.

This brings us to zero factorial. The set with zero elements is called the empty set. To find the value of zero factorial we ask, “How many ways can we order a set with no elements?” Here we need to stretch our thinking a little bit. Even though there is nothing to put in an order, there is one way to do this. Thus we have that 0! = 1.

Formulas

Another reason for the definition of 0! = 1 has to do with the formulas that we use for permutations and combinations. This is not as satisfying of an answer in explaining why zero factorial is one, but it does show why setting 0! = 1 is a good idea.

A combination is a grouping of elements of a set without regard for order. For example, we will consider the set {1, 2, 3}. There is one combination consisting of all three elements. No matter what order we arrange these elements, we end up with the same combination.

We use the formula for combinations, with the combination of three elements taken three at a time and see the following:

1 = C (3, 3) = 3!/(3! 0!)

If we treat 0! as an unknown quantity and solve algebraically, we see that 3! 0! = 3! and so 0! = 1.

Other Reasons

There are other reasons why the definition of 0! = 1 is correct, but the reasons above are the most straightforward. The overall idea in mathematics is when new ideas and definitions are constructed, they remain consistent with other mathematics. This is exactly what we see in the definition of zero factorial is equal to one.

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