Why 0 factorial is equal to 1

Add a comment. Active Oldest Votes. There are plenty of resources that already answer this question. Eric Naslund Eric Naslund However, I'm still curious why there is 1 way to permute 0 things, instead of 0 ways.

But if you say there are 0 ways to do nothing, then you are implying that it is impossible to do nothing, which is of course not the case.

This is how I look at it. Well, one. PrimeNumber PrimeNumber Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post.

Linked Related Hot Network Questions. The definition of the factorial states that 0! 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 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! 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.

A permutation is a specific, unique order of elements in a set. We could also state this fact through the equation 3! In a similar way, there are 4! So an alternate way to think about the factorial is to let n be a natural number and say that n! This corresponds to 2!

This brings us to zero factorial. The set with zero elements is called the empty set. Even though there is nothing to put in an order, there is one way to do this. Thus we have 0! Another reason for the definition of 0! This does not explain why zero factorial is one, but it does show why setting 0! A combination is a grouping of elements of a set without regard for order. No matter how we arrange these elements, we end up with the same combination.

There are other reasons why the definition of 0! 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. Actively scan device characteristics for identification. Use precise geolocation data. Select personalised content. Create a personalised content profile. Measure ad performance. Select basic ads.