Permutations with Repetitions

Topics: Permutation

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If we select $k$ objects from a set of $n$ distinct elements, and we can repeat our selection, then the number of permutations we can have is simply:

$${}_n{\text{PR}}_k=n^k$$

Example

Let’s say we are trying to calculate the amount of licence plates we can have when each plate contains $3$ letters and $4$ digits.

For the letters, considering an alphabet of 26 letters, we can have ${}_{26}{\text{PR}}_3=26^3=17576$ different permutations.

For the digits, we can have ${}_{10}{\text{PR}}_3=10^4=10000$ different permutations.

Now, since we’re choosing 3 letters and 4 digits, the number of total permutations is given by the multiplication principle:

$$\text{Total Permutations}=17576(10000)=175760000$$