OlympiadMediumCombinatorics10–12

How many permutations of $1, 2, 3, 4, 5$ leave exactly one number in its original position?

Quantas permutações de $1, 2, 3, 4, 5$ deixam exatamente um número na sua posição original?

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Solution

Choose the fixed number in $5$ ways, then derange the other four: $D(4) = 9$ . That gives $5 \times 9 = 45$ . The $44$ is $D(5)$ , the count with no fixed point at all.

How many permutations of 1,2,3,4,51, 2, 3, 4, 51,2,3,4,5 leave exactly one number in its original position?

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How many permutations of $1, 2, 3, 4, 5$ leave exactly one number in its original position?