The Forty-Sixth Annual William Lowell Putnam Competition
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Done all A–1 Determine, with proof, the number of ordered triples (A1, A2, A3) of sets which have the property that (i) A1 ∪A2 ∪A3 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and (ii) A1 ∩A2 ∩A3 = ∅. Express your answer in the form 2357, where a, b, c, d are nonnegative integers. Solution: For each integer i from 1 to 10 you must put i in one of 6 sets: A1 ∩ A2 ∩ A3 or A1 ∩ A2 ∩ A3 or A1 ∩ A2 ∩ A3 or A1 ∩ A2 ∩ A3 or A1 ∩ A2 ∩ A3 or A1 ∩ A2 ∩ A3. Thus there are 6 choices for each of the 10 values of i giving the answer 6 = 23.
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