How to Find 0 Factorial

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The factorial of 0 seems like an odd case. How do you calculate it? Read this lesson to learn what the answer is. You'll see that there are no calculations to be done and the answer is always the same.

The Steps

In this lesson, you'll quickly learn how to find 0 factorial. A factorial is a math problem that multiplies a number with all the numbers below it down to 1. It is symbolized with an exclamation mark. So 4! simply means 4 * 3 * 2 * 1.

Because the factorial has this definition, the 0! is unique. You can't multiply 0 down to 1, it's already past 1. And if you do multiply it by 1, your answer is 0. But 0 factorial does not equal 0.

To find 0 factorial, there is only one step you need to remember:

0! = 1

That is it. The answer of 0 factorial is 1. There are no calculations, nothing! All you have to do is write down 1 wherever and whenever you see 0!

But why is this? If you think about factorials as the number of possible combinations given a set of numbers, then 0 factorial means that there is only 1 possible way of combining 0 numbers. And isn't that true? If you have no numbers, then there is only one possibility: no numbers. If you have 1 factorial, then you have 1 number with only 1 possible combination. If you have 2 factorial, then you have 2 numbers with 2 possible combinations. For example, if your two numbers are 1 and 2, then you can have either {1, 2} or {2, 1} as the two possible combinations.

The Solution

Let's take a look at using this information in a problem.

Elisa is working on this math problem:

The Problem
0 factorial

She needs to find what this equals for n=0.

Elisa begins by plugging in 0 for n. She now rewrites her problem like this.

0 Factorial
0 factorial

Now, she uses the 0 factorial steps to evaluate her 0 factorial. She remembers that 0 factorial is 1 so she replaces the 0 factorial with a 1.

0! = 1
0 factorial

She can now finish by evaluating her problem. She gets 4 / 3 as her answer. So when n = 0, her problem is equal to 4 / 3.

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