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n factorial,denoted as
n!
n! = 1 * 2 * 3 * ... * (n-2) * (n-1) * n
The factorial of n can alternatively be defined as the product of all the natural numbers less than n. Therefore, it is also very common to see the factors listed in descending order.
n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1
The table below shows the factorial of some numbers. By rewriting the expressions, it can be seen that a special pattern emerges.
| 0.25cm n 0.25cm | 1cm n! 1cm | Simplify | Rewrite |
| 1 | 1 | 1 | 1 |
| 2 | 2 * 1 | 2 | 2 * 1! |
| 3 | 3 * 2 * 1 | 6 | 3 * 2! |
| 4 | 4 * 3 * 2 * 1 | 24 | 4 * 3! |
| 5 | 5 * 4 * 3 * 2 * 1 | 120 | 5 * 4! |
| ... | ... | ... | ... |
From the table above, it can be concluded that the factorial of a number follows a recursive rule.
1!=1, n! = n * (n-1)!
The definitions discussed above apply to positive integers, but it may be helpful to know the value of 0 factorial. By convention, 0!=1.
0! = 1
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.