Question

please use multinomial coefficient and explain it and follow the comment

In the first round of a knockout tournament involving n = 2m players, the n players
are divided into 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?

Answer 1

(a)

In tournament of 8 players, there will be 8/2 = 4 games in the initial round. For each game, there will be 2 outcomes (any of two teams can win)

So, possible outcomes for the initial round = 2 * 2 * 2 * 2 = 16

(b)

In initial round, possible outcomes = 16

In first round, there will be 4 teams based on the outcomes of initial round. So, in first round, there will be 4/2 = 2 games.

For each game, there will be 2 outcomes (any of two teams can win)

So, possible outcomes for the first round = 2 * 2 = 4

In second (final) round, there will be 2 teams based on the outcomes of first round. So, in final round, there will be 2/2 = 1 game.

For each game, there will be 2 outcomes (any of two teams can win)

So, possible outcomes for the final round = 2

Number of possible outcomes of the tournament = possible outcomes for the initial round * possible outcomes for the first round * possible outcomes for the final round

= 16 * 4 * 2 = 128