In the first round of a knockout tournament involving $n = 2^m$ players, the $n$ players
are divided into $\large \frac{n}{2}$ pairs, with each of these pairs then playing a game. The losers
of the games are eliminated while the winners go on to the next round, where the
process is repeated until only a single player remains. Suppose we have a knockout
tournament of 8 players.
(a) How many possible outcomes are there for the initial round? (For instance, one
outcome is that 1 beats 2, 3 beats 4, 5 beats 6, and 7 beats 8. )
(b) How many outcomes of the tournament are possible, where an outcome gives
complete information for all rounds?
