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 $VR_{n,k}$.

For 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: $ab$, $aa$, $ac$, $dc$, $cc$, $ee$, $ae$, $ea$, $bc$, $of$, $bb$, $cd$, $be$, etc...
• The variations with repetition of these 5 elements taken in threes are: $abc$, $abb$, $acd$, $ccc$, $aba$, $dce$, $eed$, $cda$, etc...
• The variations with repetition of these 5 elements taken in fours are: $abbd$, $acdd$, $beac$, $eecc$, $dace$, etc...
• The variations with repetition of these 5 elements taken in fives are: $abcde$, $abbbc$, $aeded$, $daece$, $bcced$, $edcba$, 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:$$VR_{n,k}=n^k$$

In the previous example,

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

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