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Question

At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives 5th prize and the winner bowls #3 in another game. The loser of this game receives 4th prize and the winner bowls #2. The loser of this game receives 3rd prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes?

Accepted Answer

We have 2 choices for who wins the first game, and that uniquely determines 5^th place. Then there are 2 choices for a next game and that uniquely determines 4^th place, followed by 2 choices for the next game that uniquely determines 3^rd place. Finally, there are 2 choices for the last game, which uniquely determines both 1^st and 2^nd places, since the winner is 1^st and the loser is 2^nd. Thus the number of possible orders is 2 2 2 2 = 16, which is 16.