Exercise 986Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

Permutations with at least one fixed point

#IE#fixed point#permutations

Statement

How many permutations of $\{1,\ldots,n\}$ have at least one fixed point?

Hint

Complement of derangements: $n!-D_n$ .

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Total. $n!$ permutations.

Derangements. $D_n$ (no fixed point).

At least one. $n!-D_n$ .

Conclusion. $n!-D_n$ permutations have at least one fixed point.

Values. $n=4$ : $24-9=15$ .