Let R be the space of all the N! permutations of (1,2,...N), and let R = (R1, ...Rn) be uniformly distributed over R. Prove that (i) P(Ri=k) = 1/N; 1<=i, k<=N. (ii)P(Ri=k, Rj=h) = 1/N(N-1), 1<= i ≠j, k≠h<=N (iii)P(Ri=k, Rj=k) = 0, 1<=i≠j, k<=N.

Aschool allots ? 1500 to spend on a trip to the theatre. theatre tickets have a regular cost of ? 55 each and are on offer for 1 5 off. a train ticket for the day will cost ? 12 each. if 2 teachers and the maximum number of students attend, how much money will the school have left over?