A cute problem on permutations and combinations

Here, is a problem on counting. Read it and crack it before you compare your solution with the one I present !!!

In a tennis tournament, there are 2n participants. In the  first round of  the tournament, each participant plays just once, so there are n games, each occupying a pair of  players. Show that the pairing for the first round can be arranged in exactly

1 x 3 x 5 x 7 x 9 \ldots(2n-1)

different ways.

Solution.

Call the required number of pairings of 2n players P_{n}. If you are a participant, you can be matched with any one of  the other (2n-1) players. Once your antagonist is chosen, there remain

(2n-2)=2(n-1)

players who  can be paired in P_{n-1} ways. Hence,

P_{n}=(2n-1)P_{n-1}.

Did you like it? Please send your solutions, comments, etc.

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Nalin Pithwa