Question

In a chess tournament, n men and 2 women players participated. Each player plays 2 games against every other player. Also, the total number of games played by the men among themselves exceeded by 66 the number of games that the men played against the women. Then the total number of players in the tournament is
1. 13
2. 11
3. 9
4. 7

Answer 1

Correct Answer - Option 1 : 13

CONCEPT:

The combination formula is used to find the number of ways of selecting items from a collection such that the order of selection does not matter.

Combination formula can be written as \(^n{C_r} = \frac{{n!}}{{r!\left( {n - r} \right)!}}\).

CALCULATIONS:

Let there be n men participants. 
Then the number of games that the men play between themselves is 2× ⁿ C _r

games that the men played with the women is $2\times (2n)$.

\(\therefore 2 \times {C_2}^n - 2 \times 2n = 66\) (Given)

Or n² - 5n - 66 = 0

On solving we get n = 11.

Hence, the number of participants is $11$ men + $2$ women $=13$.

So, option (1) is the correct answer.