Permutations with no Repetitions

Topics: Permutation

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If we select $k$ objects from a set of $n$ distinct elements, and we cannot repeat our selection, then the number of permutations we can have is:

$${}_n{\text{P}}_k=\frac{n!}{(n-k)!}$$

For a set with subsets of equal elements, see permutations with groups of equal elements.

Example

How many permutations can be made with the letters of the word “COLUMN”?

Since “COLUMN” has 6 letters and it contains no repeated letters, then we can have ${}_6{\text{P}}_6=6!=720$ permutations.

How many 4-letter permutations can be made with the letters of the word “COLUMN”?

This time, we can have ${}_6{\text{P}}_6=\frac{6!}{(6-4)!}=\frac{6!}{2!}=6(5)4(3)=360$ permutations.