CSAT Solved Papers/ 2022/Q17

2022 CSAT — Q17

Quant Counting & combinatorics 2.5 marks Easy

In a tournament of Chess having 150150 entrants, a player is eliminated whenever he loses a match. It is given that no match results in a tie/draw. How many matches are played in the entire tournament?

  1. A 151
  2. B 150
  3. C 149 Answer
  4. D 148

Worked rationale

Count by eliminations, not brackets. Every match produces exactly one loser, and a player is eliminated the moment they lose. To crown a single champion, all other 149149 players must be eliminated — one per match.

matches=150−1=149.\text{matches} = 150 - 1 = 149.

This holds regardless of byes or how the bracket is seeded.

Answer: (c) 149.

Why the other options miss

  • A
    one match too many: adds matches for a phantom final, or counts the champion as still needing a match.
  • B
    one per player, not per loss: counts one match per player rather than one per eliminated player.
  • D
    one elimination short: stops at 148148 eliminations, leaving two players “uncrowned.”

Specialist insight

The elegant invariant: one match ⇔\Leftrightarrow one elimination. With 150150 entrants and one winner, exactly 149149 players must be eliminated, hence 149149 matches — no bracket diagram, no powers of two needed. This “loser-counting” argument is bye-proof and is the fastest correct route under the clock.

The trap, in one line

One loser per match, 149149 players must be eliminated ⇒149\Rightarrow 149 matches ⇒\Rightarrow (c).

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