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August 4th, 2017, 02:25 PM   #1
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How Many Outcomes Could This Game Have?

A game has four players and three rounds. In each round, the last place player doesn't get any points. In Round 1, first place gets 3 points, second place gets 2 points, and third place gets 1 point. Round 2 has double the points of Round 1 and Round 3 has triple the points of Round 1. There are 36 available points. Given that the players got 13, 10, 9, and 4 points, how many outcomes could have produced that score distribution? Player 1 having 13 points and Player 2 having 10 points is the same as Player 1 having 10 points and Player 2 having 13 points. For starters, the player with 13 points must have gotten at least 6 points from Round 3 because Round 1 and Round 2 combined can't give a player more than 9 points and 9 + 3 < 13. The player with 4 points could not have finished first in Round 2 or first or second in Round 3.

Round 1: 3 + 2 + 1 + 0 = 6
Round 2: 6 + 4 + 2 + 0 = 12
Round 3: 9 + 6 + 3 + 0 = 18

There are 4^3 = 64 possibilities, so I could do the problem by hand if I had to.
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