# 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,

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