Basic Combinatorics

Table of contents

Permutation

A permutation is an arrangement of objects in a specific order.

If repetition is not allowed, then the number of permutations of $n$ objects taken $r$ at a time is

$_{n} P_{r} = \frac{n!}{(n - r)!}$

If repetition is allowed, then the number of permutations of $n$ objects taken $r$ at a time is $n^{r}$ .

A combination is a selection of objects where order does not matter.

If repetition is not allowed, then the number of combinations of $n$ objects taken $r$ at a time is

$_{n} C_{r} = \frac{n!}{r! (n - r)!}$

If repetition is allowed, then the number of combinations of $n$ objects taken $r$ at a time is

$_{r + n - 1} C_{r} = \frac{(r + n - 1)!}{r! (n - 1)!}$