Exercise 1723Combinatorics › Inclusion-exclusionOlympiadAdded 3 Nov 2026

Combining multiple IE applications

#IE#combining#multi-step

Statement

Count permutations of $\{1,\ldots,n\}$ where both $1$ and $2$ are fixed.

Hint

Fix both, count permutations of the remaining $n-2$ .

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Fix 1 and 2. Then count permutations of $\{3,\ldots,n\}$ : $(n-2)!$ .

Conclusion. $(n-2)!$ permutations fix both $1$ and $2$ .

IE interpretation. $|A_1\cap A_2|=(n-2)!$ .