Factorial

This may seem strange since, by the definition of factorial, this means that the product of multiplying no numbers is equal to 1. Nevertheless, defining 0! as 1 is necessary for the usage of the factorial of a number. For example, if the recursive property is used with n= 1, the following result is obtained. n! &= n * (n-1)! [1em] 1! &= 1 * ( 1-1)! 1! &= 1* 0! 1 &= 1* 0! The last equality, 1 = 1* 0!, only holds true if 0!=1. In a similar way, there are many applications of the factorial in different areas of mathematics that only make sense if 0!=1. For example, the number of ways to choose groups of r elements from a set of n elements — combinations of n in r — is given by the formula shown below. _nC_r = n!/(n-r)!n! In particular, there is only one way to make groups of n elements from a set having n elements. This is by making a group using all the elements. This is what happens when substituting n= n and r= n. _nC_n &= n!/( n- n)! n! [0.75em] 1 &= n!/0! n! [0.75em] 1 &= 1/0! Once more, for everything to be consistent, the only value that can be assigned to 0! is 1.