 # Using the fact that n! gives the number of permutations of a set of n elements, provide another proof of P(A) obtained at (2)-(f) and generalize it to the case of n teams (n being even).

how many possible draws feature the game Republic of Ireland vs Germany? (f) Derive the probability of the event A={the Republic of Ireland will avoid Germany at this stage of the tournament}. Was there a more direct way to derive this result? [15 pts] (g) [Bonus Question] Another way to derive the result of (2)-(f). Denote by T1, T2, . . . , T16 the 16 teams involved in the tournament. An efficient method to simulate a random draw of 8 games from a computer would be: Step 1: shuffle the order of the elements of the vector v = [T1 T2 · · · T16] i.e. generate a random permutation σ(v) of the vector v. For example, the generated permutation might be σ(v) = [T13 T6 T4 T10 · · · T11 T1] . Step 2: split the vector by groups of two consecutive elements: in the case of the permutation generated above σ(v), define the games as T13 vs T6, T4 vs T10, …, T11 vs T1. Here, we consider that the order of the games does not matter. More precisely, the two following draws are regarded as identical: T13 vs T6 T4 vs T10 T2 vs T5 T16 vs T12 T3 vs T15 T9 vs T7 T14 vs T8 T11 vs T1 T4 vs T10 T13 vs T6 T2 vs T5 T16 vs T12 T3 vs T15 T9 vs T7 T14 vs T8 T11 vs T1 Using the fact that n! gives the number of permutations of a set of n elements, provide another proof of P(A) obtained at (2)-(f) and generalize it to the case of n teams (n being even).