Which of the statements given is/are correct?

Question

In a carrom board game competition, m boys 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:
I. The total number of students that participated in the competition is 30.
II. The number of games in which both players were girls is 78.

Which of the statements given above is/are correct?
(a) I only (b) II only
(c) Both I and II (d) Neither I nor II.

Answer 1

C. Both statements are correct.
.
For statement 2, if 78 games had 2 female players, then there must be 13 girls
(number of games = 0+1+2+3+4+5+6+7+8+9+10+11+12=78
With 13 girls, there need to be 221/13 = 17 boys, which exceeds 13 girls.
Statement 2 is correct.
.
For statement 1, with the 13 girls and 17 boys in statement 2 the total number of players - 30.
Statement 1 is correct.