In a carrom board game competition, m boys and n girls (m > n > 1) of a school participate in which every student has to play exactly one game with every other student. Out of the total games played, it was found that in 221 games one player was a boy and

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Q: 121 (IAS/2009)
In a carrom board game competition, m boys and n girls (m > n > 1) of a school participate in which every student has to play exactly one game with every other student. Out of the total games played, it was found that in 221 games one player was a boy and the other-player was a girl. Consider the following statements :
1. The total number of students that participated in the competition is 30.
2. The number of games in which both players were girls is 78.
Which of the statements given above is/are correct ?

question_subject: 

Logic/Reasoning

question_exam: 

IAS

stats: 

0,6,1,0,1,6,0

keywords: 

{'carrom board game competition': [0, 0, 1, 0], 'total games': [0, 0, 1, 0], 'competition': [1, 0, 1, 2], 'school participate': [0, 0, 1, 0], 'students': [0, 1, 1, 1], 'players': [0, 1, 4, 2], 'games': [0, 0, 9, 4], 'other student': [0, 0, 1, 0], 'total number': [0, 0, 3, 0], 'game': [0, 0, 4, 1], 'student': [0, 0, 2, 4], 'player': [1, 0, 1, 2], 'girls': [0, 2, 3, 10], 'number': [0, 0, 0, 2]}

This question involves combinatorics for understanding the number of games played between boys and girls, and the number of games played only among the girls during a carrom competition.

The total number of games played between boys and girls is given by the product of `m` and `n`; This number is set to be 221 for this question.

Statement 1 suggests that the total students (boys and girls combined) that participated in the competition is 30, we`ll confirm this after know the number of boys and girls.

Statement 2 claims that the total number of games played among girls themselves is 78. This number is given by the combination of `n` girls taken 2 at a time, represented by the mathematical formula nC2. We will confirm this once we have the value for `n`.

Upon solving these equation, we would indeed find that `m` is 16 (representing boys) and `n` is 14 (representing girls), which validate both the statements. Thus, the total number of students is indeed 30 (16+14), and the number of games played among the girls is indeed 78.

Therefore, both statements 1 and 2 are correct, making option 3 the right answer.