Next month I'll be taking part in the annual KCLMS tiddlywinks tournament. There will be 128 players, and the tournament structure is single elimination. This means that there are 64 matches in the first round, and the winners go on to the second round (while the losers are eliminated). There are then 32 matches in the second round, and so on - with the final round being a match between the last two players.

Problem 1: how many different arrangements of players are possible in the first round?

(To clarify, let's take a four-person tournament with players A, B, C, D. (A v B, C v D), (A v C, B v D), (A v D, B v C) are the possible first round match-ups, so there are 3 possible first rounds.)

Problem 2: my nemesis, Dr Ironfist, will be playing in the tournament. What is the probability that I will play her at some point in the tournament? (Assume that tiddlywinks is a game of chance, and that pairings for each round are generated randomly from the players who progressed to that round.)