linguistic:voice_change
Definition

The factorial function of a positive integer n {\displaystyle n} is defined by the product of all positive integers not greater than n {\displaystyle n} n ! = 1 ⋅ 2 ⋅ 3 ⋯ (n − 2) ⋅ (n − 1) ⋅ n. {\displaystyle n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n.} This may be written more concisely in product notation as n ! = ∏ i = 1 n i. {\displaystyle n!=\prod _{i=1}^{n}i.}

If this product formula is changed to keep all but the last term, it would define a product of the same form, for a smaller factorial. This leads to a recurrence relation, according to which each value of the factorial function can be obtained by multiplying the previous value by n {\displaystyle n}: n ! = n ⋅ (n − 1) !. {\displaystyle n!=n\cdot (n-1)!.} For example, 5 ! = 5 ⋅ 4 ! = 5 ⋅ 24 = 120 {\displaystyle 5!=5\cdot 4!=5\cdot 24=120}.