Permutations — Meaning and Formula (nPr)

Meaning of a permutation

A permutation is an arrangement of a group of objects in a definite ORDER. Placing the same 3 people into the roles of President, Secretary and Treasurer in a different order gives three genuinely different outcomes — that is the hallmark of a permutation problem: order matters.

Building the formula from first principles

Suppose $r$ objects, chosen out of $n$ distinct objects, are placed into $r$ positions in a row:

• The 1st position can be filled in $n$ ways (any of the $n$ objects).
• The 2nd position can then be filled in $(n-1)$ ways (one object is already used up).
• The 3rd position can be filled in $(n-2)$ ways.
• … continuing, the $r$-th position can be filled in $(n-r+1)$ ways.

By the counting principle, the total number of arrangements is the product of these descending factors:

Multiplying and dividing by $(n-r)!$ turns this into the compact factorial form used throughout this chapter:

Evaluating $^{n}P_{r}$

To evaluate $^{n}P_{r}$, either apply the formula directly, or — usually faster — write out just the first $r$ descending factors of $n!$: …