The number of bijections f(·) from the set S = \1, 2, 3, 4\ to itself such that f(f(n)) = n, for all n ∈ S, is __________. (Answer in integer)

GATE DA 2026 Q33: Number of involutions on a 4-element set

Probability and Statistics
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The number of bijections $f(\cdot)$ from the set $S = \{1, 2, 3, 4\}$ to itself such that $f(f(n)) = n$ , for all $n \in S$ , is __________. (Answer in integer)

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Answer: 10 (official GATE 2026 final answer key)

Concept used: A bijection with $f(f(n)) = n$ (an involution) consists only of fixed points and swapped pairs.

First step: Count by number of swapped pairs: none (1 way), one pair ( $\binom{4}{2} = 6$ ), two pairs (3).

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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GATE DA 2026 Q33: Number of involutions on a 4-element set

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