GATE 2026 DA – Question 33
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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Correct answer: 10
Explanation
A bijection with $f(f(n)) = n$ is its own inverse (an involution), so it is made only of fixed points and swaps of two elements. The identity has no swaps (1 way). With one swap there are $\binom{4}{2} = 6$ ways. With two swaps there are 3 ways of pairing up the four elements. The total is $1 + 6 + 3 = 10$.