GATE DA 2026 Set 1 — Question 33
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Probability & Statistics → Counting → Permutations
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Question
The number of bijections from the set to itself such that , for all , is __________ . (Answer in integer)
Correct answer
10 to 10
Solution
The condition implies that the function is an involution. In terms of cycle decomposition, an involution consists only of cycles of length 1 (fixed points) or length 2 (transpositions).For a set of size 4, the possible cycle structures are:
1.Four fixed points (1-cycles):
- This is the identity permutation.
- Number of ways = 1.
- Choose 2 elements for the transposition: .
- The remaining 2 elements are fixed points.
- Number of ways = 6.
- Choose 2 elements for the first pair: .
- The remaining 2 form the second pair.
- Since the order of disjoint transpositions does not matter (e.g., is the same as ), we divide by 2.
- Number of ways = .
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