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For any integer which is positive, the product of all integers from to is called factorial, denoted as

The factorial of can alternatively be defined as the product of all the natural numbers less than Therefore, it is also very common to see the factors listed in descending order.

The table below shows the factorial of some numbers. By rewriting the expressions, it can be seen that a special pattern emerges.

Simplify Rewrite

From the table above, it can be concluded that the factorial of a number follows a recursive rule.

The definitions discussed above apply to positive integers, but it may be helpful to know the value of factorial. By convention,


Why is defined to be ?

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