How many one-to-one functions are there from a set A with n elements onto itself?

Question

Accepted Answer

There are n! one-to-one onto (bijective) functions from an n-element set to itself. A one-to-one function from a finite set onto itself is a bijection, i.e. a permutation of the n elements. To build one we decide where each element maps, with no two elements allowed to share an image. The first element has n possible images, the next has n-1 remaining choices, then n-2, and so on down to 1 choice for the last. Multiplying these gives n x (n-1) x ... x 1 = n! distinct bijections.