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Interview prep / Probability

Distinct sums of three products from shuffling 1 to 6

Question

The numbers 1,2,3,4,5,61,2,3,4,5,6 are placed in a random order. The first two are multiplied together, as are the middle two and the last two, and the three products are added. Over all possible orders, how many different values can this sum take?

Short answer

99. Only the split into three unordered pairs matters, and there are 5⋅3⋅1=155\cdot3\cdot1=15 such pairings. Listing their sums gives just nine distinct values: 28,29,31,32,35,37,40,41,4428,29,31,32,35,37,40,41,44 (several pairings collide, e.g. 1⋅2+3⋅6+4⋅5=1⋅4+2⋅3+5⋅6=401\cdot2+3\cdot6+4\cdot5=1\cdot4+2\cdot3+5\cdot6=40).

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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