Question

Saga Internationals have conducted a chess competition between young boys and girls, wherein every individual has to play exactly one game with every other individual. It was found that in 66 games, both the players were girls, and in 210 games, both were boys. Find the number of games in which one player was a boy and the other was a girl.

• A. 210
• B. 222
• C. 252
• D. 276
• E. 290

Answer 1

The Correct Option is C

To solve this problem, we need to determine the number of games where one player was a boy and the other was a girl. Let's break it down step-by-step:

1. Let \(G\) be the number of girls and \(B\) be the number of boys. 
2. According to the problem:
   • The number of games where both players are girls is 66. This forms a combination of 2 girls playing together, represented by \(\binom{G}{2} = 66\).
   • The number of games where both players are boys is 210. This forms a combination of 2 boys playing together, represented by \(\binom{B}{2} = 210\).
3. We use the combination formula: \(\binom{n}{2} = \frac{n(n-1)}{2}\).
   • From \(\binom{G}{2} = 66\): \(\frac{G(G-1)}{2} = 66\) so, \(G(G-1) = 132\).
   • From \(\binom{B}{2} = 210\): \(\frac{B(B-1)}{2} = 210\) so, \(B(B-1) = 420\).
4. Solving these quadratic equations:
   • Solving \(G(G-1) = 132\):
     • Try \(G = 12\): \(12 \cdot 11 = 132\) (True)
   • Solving \(B(B-1) = 420\):
     • Try \(B = 21\): \(21 \cdot 20 = 420\) (True)
5. Now, the total number of games where one player is a boy and the other is a girl can be calculated as: \(G \times B = 12 \times 21 = 252\).

Thus, the number of games where one player was a boy and the other was a girl is 252. This corresponds to the correct answer:

• 252