**From New Scientist #2498, 7th May 2005**

Of the 8 fifth round games in this year’s FA cup competition 4 were won and 4 required replays. When the draw for the quarter-finals was made each game matched a team that had won against the winner of a replay.

(1) What as a fraction expressed in its lowest terms was the probability that the draw would produce this outcome?

(2) What was the probability (similarly expressed) that it would produce two games that involved the four teams that had won and two that involved the winners of the four replays?

[enigma1339]

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*Related*

This is a bit similar to

Enigma 1344, in that they are both about pairing up members of distinct sets. This Python program constructs all parings (using thepartitions()function fromenigma.py, originally created forEnigma 1449but useful in several puzzles) and counts how many satisfy the conditions for each part of the problem. It runs in 54msSolution:(1) 8/35; (2) 3/35.Here is a program that determines the solution analytically. I used the

pairs()function from my solution to Enigma 1344. It agrees with the solution produced by the constructive approach and runs in 51ms.This enigma is much easier and quicker to do just by analysis.