@Mircea: I'm not sure that is correct either.
Let's try N=3, player i guesses i all the time. The contributions of player 1 (games where he is the only one correct) will be: (1, 1, 1), (1, 1, 2), (1, 3, 1), (1, 3, 2). N^(N-2) = 3 (not 4).

Also, once you get the right solution it will fit in 300 chars of layman's terms.

@Mircea
Your reasoning for Part2 is not correct. Let's take N=2, player 1 can't see anything so he just guesses 1 all the time, out of the 4 possibilities he will be right 2 times ,  N^(N-1) - N^(N-2) = 2-1 = 1