The chess clubs of two schools compete against each other every year. Each school team has n players and on the day of the contest, each member of the first team is drawn to play against a member of the second team. In two consecutive contests between the two teams, each of the two teams has the same members and two separate draws, one in each occasion, take place. We want to calculate the probability of the event Bn∶ at least one member of the first team plays against the same opponent of the second team in the two contests.

Let us define the events Ai∶ the ith member of the first team plays against the same opponent in the two contests for i = 1, 2,…, n.
(i) For the special case n = 3, verify that

(ii) Using the Poincaré formula (see Proposition 1.10), show that, in the general case,

(iii) Verify that, as n → ∞, we have

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and conclude that P(A) = P(A) = P(A3) = 2! 3!' 1! P(AA) = P(AA3) = P(A2A3)= | P(AAA3) 3!" P(B3) = 1 - 2! 3!