Permutations

A permutation is an ordered arrangement of objects. Suppose three friends A, B, C must stand in a line for a photograph — some possible orderings (left to right) are A,B,C; A,C,B; B,A,C; B,C,A; C,B,A; C,A,B, giving six possible arrangements in total.

This matches $3\times2\times1=3!$: filling the first position has 3 choices, the second has 2 remaining choices, the third has the 1 leftover choice (rule of product). In general, $n$ distinct objects arranged in a row have $n!$ possible permutations.

Choosing and arranging only some of the objects. Suppose we have 7 letters A–G and want to build a 4-letter string (no repeats): the 1st letter has 7 choices, the 2nd has 6, the 3rd has 5, the 4th has 4, giving $7\times6\times5\times4$ strings. Writing this as a ratio of factorials,

$7\times6\times5\times4 = \frac{7\times6\times5\times4\times3\times2\times1}{3\times2\times1} = \frac{7!}{3!} = \frac{7!}{(7-4)!}.$ …