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Let sets A and B satisfy n(A) = 4 and n(B) = 3. The ordered pairs (x, 1), (y, 2), (z, 1), and (2, 3) all belong to A x B, where x, y, z are distinct elements. If lambda denotes the number of onto (surjective) functions from A to B, find lambda / 6.

1. 6
2. 10
3. 12
4. 36

Correct answer: 36

Solution

From the given pairs, second components are from B: B = {1, 2, 3}. First components are from A: x, y, z, 2 (four distinct elements, so A = {x, y, z, 2}). Number of onto functions from A (4 elements) to B (3 elements) by inclusion-exclusion: lambda = 3⁴ - C(3,1)*2⁴ + C(3,2)*1⁴ = 81 - 3*16 + 3*1 = 81 - 48 + 3 = 36. Therefore lambda/6 = 36/6 = 6.

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