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A mapping is chosen at random from all mappings of the set S = {1, 2,..., n} to itself. If the probability that the chosen mapping is one-one (bijective) equals 5/324, find the value of n.
- 4
- 5
- 6
- 3
Correct answer: 6
Solution
Total number of mappings from an n-element set to itself = nⁿ (each element has n choices). Number of one-one (bijective) mappings = n! (permutations). Probability = n!/nⁿ. We need n!/nⁿ = 5/324. Testing n = 6: 6! = 720, 6⁶ = 46656. 720/46656 = 720/46656. Simplify: GCD(720, 46656). 46656/720 = 64.8, so 720*64 = 46080, 46656 - 46080 = 576. GCD(720, 576) = 144. 720/144 = 5, 46656/144 = 324. So 6!/6⁶ = 5/324. Confirmed.
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