Question

There are two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played among themselves proved to be exceeded by $ 66 $ number of games that the men played with the women. The number of participants is
(a) $ 6 $
(b) $ 11 $
(c) $ 13 $
(d) None of these

Answer 1

Hint: Start by mentioning all the given conditions. Then evaluate each and every condition and according to that evaluate the values. After evaluating the values for each condition, put all the terms together and evaluate the final answer.

Complete step by step solution:
First we will start off by considering the number of men participating in tournaments as $ n $ and we know that the number of women participating in tournaments are $ 2 $ . Now, here we know that each player will play $ 2 $ games with every player.
Hence, two women will play two games with each man. Therefore, total games played by women will be $ 2 \times 2n $ , which will be equal to $ 4n $ .
Therefore, total number of games played by a man among themselves will be given by,
(Total number of pairs of possible men) $ \times 2 $ each pair will play.
So, now we can write $ n(n - 1) - 4n = 66 $ .
Now we will solve for the value of $ n $ .
 $
  n(n - 1) - 4n = 66 \\
  {n^2} - 5n - 66 = 0 \\
  {n^2} - 11n + 6n - 66 = 0 \\
  (n - 11)(n + 6) = 0 \\
  n = 11, - 6 \;
  $
But the value of $ n $ cannot be zero. Hence, the value of $ n $ will be $ 11 $ .
Now the total number of participants will be the total of man and woman participants.
 $
   = 11 + 2 \\
   = 13 \;
  $
Hence, the total games played will be the total number of players $ \times 2 $ .
 $
   = {}^{13}{C_2} \times 2 \\
   = \dfrac{{13 \times 12}}{2} \times 2 \\
   = 156 \;
  $
Hence, the total number of participants is $ 13 $ that is option (C).
So, the correct answer is “Option C”.

Note: While mentioning conditions, make sure to mark all the important words so that you do not miss any important terms. Then next always evaluate all the conditions and then evaluate their values step by step to avoid any confusions.