# Variation

Keywords: variation without repetition with repetition

## Variation without Repetition

Variation is a way of selecting k items from a collection of n items (k ≤ n), such that (like permutations) the order of selection does matter. The repetition of items is not allowed.

The number of variations::

${V}_{n}^{k}=\frac{n!}{\left(n-k\right)!}$

Example:

From 4 items {a,b,c,d,} choose 2, repetition is not allowed:${V}_{4}^{2}=\frac{4!}{\left(4-2\right)!}=12$

(a,b), (a,c), (a,d), (b,c), (b,d), (c,d), (b,a), (c,a), (d,a), (c,b), (d,b), (d,c)

## Variation with Repetition

A combination is a way of selecting k items from a collection of n items, such that (like permutations) the order of selection does matter. The repetition of items is allowed.

The number of variations:

${\overline{V}}_{n}^{k}={n}^{k}$

Example:

From 4 items {a,b,c,d} choose 2 items, repetition is allowed:

The number of variations:: ${\overline{V}}_{4}^{2}={4}^{2}=16$

(a,a), (b,a), (c,a), (d,a), (a,b), (b,b), (c,b), (d,b), (a,c), (b,c), (c,c), (d,c), (a,d), (b,d), (c,d), (d,d)