Proof of why 0! Is Equal to 1!

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Proof of why 0! Is Equal to 1!


If you are a high school student taking probability and statistics, you would have encountered the topic of permutations and combinations. Within this topic, the use of factorials are prevalent. A factorial – represented by ‘!’ or exclamation mark- is a function that represents a descending multiplication of a number until 1. For example, 4! is 4 x 3 x 2 x 1 = 24. However,  there is something called ‘0!’. According to the principal established above, a factorial descends until 1 but 0 is less than 1. Yet, it is taught that the answer is 1. At first, this seems illogical, but if the perspective of how the factorial is defined is changed, 0! does, in fact, equate to 1.

Look at the factorial table here:


As the factorials (n!) move up the table, the previous term ((n-1)!) is multiplied by n. This is the conventional representation of multiplication. However, what if the perspective of division is taken?

As the magnitude (n) of the factorial decreases, the term is divided by n, leading to descending division. When 0! is reached, 1! is divided by 1, leading to 0! = 1 = 1!.

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