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!