Variations with repetition

Where $A$ is a set of $n$ elements. The variations with repetition of $n$ elements taken $k$ by $k$ are the arranged groups formed by $k$ elements from $A$ (which may be repeated). This is represented as $V R_{n, k}$ .

For example,

Example

If the set of $5$ elements is $A = {a, b, c, d, e}$ :

The variations with repetition of these 5 elements taken in ones are: $a$ , $b$ , $c$ , $d$ and $e$ .
The variations with repetition of these 5 elements taken in twos are: $a b$ , $a a$ , $a c$ , $d c$ , $c c$ , $e e$ , $a e$ , $e a$ , $b c$ , $o f$ , $b b$ , $c d$ , $b e$ , etc...
The variations with repetition of these 5 elements taken in threes are: $a b c$ , $a b b$ , $a c d$ , $c c c$ , $a b a$ , $d c e$ , $e e d$ , $c d a$ , etc...
The variations with repetition of these 5 elements taken in fours are: $a b b d$ , $a c d d$ , $b e a c$ , $e e c c$ , $d a c e$ , etc...
The variations with repetition of these 5 elements taken in fives are: $a b c d e$ , $a b b b c$ , $a e d e d$ , $d a e c e$ , $b c c e d$ , $e d c b a$ , etc...

The following formula gives us a much quicker way of counting all the variations with repetition of $n$ elements taken $k$ by $k$ . There is: $V R_{n, k} = n^{k}$

In the previous example,

Example

The number of variations with repetition of 5 elements of A taken in threes is: $V R_{5, 3} = 5^{3} = 5 \cdot 5 \cdot 5 = 125$

It is visibly much more practical to use the formula than to try all the possibilities by hand!