In a darts tournament two groups of 16 players each are drawn against each other and the winners go forward to the next round. Two players fail to turn up and their opponents are given an automatic win. What is the total number of matches that will have to be played to a winner?

Question

A 29
B 30
C 31
D 32

Please explain. Much thanks

Answer 1

With 16 players , number of total games
= C(16,2) = 120
with only 14 players, number of total games
= C(14,2) = 91

120-91= 29

The wording of this question is not clear.
In a typical "knock-out" tournament, once you lose a game you are done
e.g. the current Australian Open

But you seem to suggest that every team plays every other team, (like
the typical "shake-hands" problem. That is why I used the C(n,r) approach
But then there would be just 91 games with the person with most wins wins.
In a 16 player tournament , there would be 8 games in round 1
4 games in round2, 2 games and finally the championship game.
Total number of games = 8+4+2+1 = 15
Such tournaments usually have number of entries that are powers of 2,
e.g. Australia has 128 in the Men's , 128 = 2^7

The ball is in your court.