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Set X has exactly 5 elements and set Y has exactly 7 elements. Let beta be the number of one-to-one (injective) functions from X to Y, and let alpha be the number of onto (surjective) functions from Y to X. Compute (1/5!) * (alpha - beta).
- 117
- 118
- 119
- 120
Correct answer: 119
Solution
beta = 7*6*5*4*3 = 2520. For alpha (onto functions from Y to X with |Y|=7, |X|=5): alpha = sumₖ₌₀⁵ (-1)^k * C(5,k) * (5-k)⁷. Compute and subtract: (alpha-beta)/5! = 119.
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