Exercise 1623Combinatorics › Counting principlesL1Added 3 Nov 2026

Permutations with fixed points

#fixed points#permutations#inclusion-exclusion

Statement

Count permutations of $\{1,\ldots,n\}$ that fix at least one element.

Hint

Complement of derangements: $n!-D_n$ .

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Solution

Complement. Permutations with at least one fixed point = total $-$ derangements:
$n!-D_n=n!-n!\sum_{k=0}^n\frac{(-1)^k}{k!}=n!\left(1-\sum_{k=0}^n\frac{(-1)^k}{k!}\right).$

Simplification. $=n!\sum_{k=1}^n\frac{(-1)^{k+1}}{k!}$ .

Conclusion. Number of permutations with at least one fixed point $=n!-D_n$ .

Permutations with fixed points

Statement

Solution

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