Why Does Zero Factorial Equal One?

A zero factorial is a mathematical expression for the number of ways to arrange a data set with no values in it, which equals one. In general, the factorial of a number is a shorthand way to write a multiplication expression wherein the number is multiplied by each number less than it but greater than zero. 4! = 24, for example, is the same as writing 4 x 3 x 2 x 1 = 24, but one uses an exclamation mark to the right of the factorial number (four) to express the same equation.

It is pretty clear from these examples how to calculate the factorial of any whole number greater than or equal to one, but why is the value of zero factorial one despite the mathematical rule that anything multiplied by zero is equal to zero? 

The definition of the factorial states that 0! = 1. This typically confuses people the first time that they see this equation, but we will see in the below examples why this makes sense when you look at the definition, permutations of, and formulas for the zero factorial.

The Definition of a Zero Factorial

The first reason why zero factorial is equal to one is that this is what the definition says it should be, which is a mathematically correct explanation (if a somewhat unsatisfying one). Still, one must remember that the definition of a factorial is the product of all integers equal to or less in value to the original number—in other words, a factorial is the number of combinations possible with numbers less than or equal to that number.

Because zero has no numbers less than it but is still in and of itself a number, there is but one possible combination of how that data set can be arranged: it cannot. This still counts as a way of arranging it, so by definition, a zero factorial is equal to one, just as 1! is equal to one because there is only a single possible arrangement of this data set.

For a better understanding of how this makes sense mathematically, it's important to note that factorials like these are used to determine possible orders of information in a sequence, also known as permutations, which can be useful in understanding that even though there are no values in an empty or zero set, there is still one way that set is arranged. 

Permutations and Factorials

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

• 1, 2, 3
• 1, 3, 2
• 2, 3, 1
• 2, 1, 3
• 3, 2, 1
• 3, 1, 2

We could also state this fact through the equation 3! = 6, which is a factorial representation of the full set of permutations. 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 0! = 1.

Formulas and Other Validations

Another reason for the definition of 0! = 1 has to do with the formulas that we use for permutations and combinations. This does not explain 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, consider the set {1, 2, 3}, wherein there is one combination consisting of all three elements. No matter how 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 that 1 = C (3, 3) = 3!/(3! 0!), and if we treat 0! as an unknown quantity and solve algebraically, we see that 3! 0! = 3! and so 0! = 1.

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 that when new ideas and definitions are constructed, they remain consistent with other mathematics, and this is exactly what we see in the definition of zero factorial is equal to one.