UPSC Prelims 2009 Q106: Science & Technology – Quantitative Aptitude
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 ?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Answer: (c) Both 1 and 2
Explanation
A boy-girl tournament combinatorics puzzle
With m boys and n girls (m > n > 1), every pair of students plays exactly one game, and 221 games were boy-vs-girl matchups.
| Step | Reasoning |
|---|---|
| Boy-girl games formula | Number of boy-vs-girl games = m × n (each boy plays each girl once) |
| Factor 221 | 221 = 13 × 17, and since m > n, we get m = 17, n = 13 |
| Statement 1: total students = 30 | 17 + 13 = 30 — Correct |
| Statement 2: girl-vs-girl games = 78 | Number of girl-girl games = C(13, 2) = 13×12/2 = 78 — Correct |
Verifying both statements:
| Point | Detail |
|---|---|
| m, n values | m=17, n=13 (from factoring 221) |
| Total students | 17+13 = 30 |
| Girl-girl games | C(13,2) = 78 |
Answer: (c) Both 1 and 2.
Why each option is right or wrong
- (a) Incorrect. Statement 2 (78 girl-girl games) is also correct, so '1 only' is incomplete.
- (b) Incorrect. Statement 1 (30 total students) is also correct, so '2 only' is incomplete.
- (c) CORRECT. With m=17 boys and n=13 girls (from factoring 221=13×17), both the total-students (30) and girl-girl-games (78) figures check out.
- (d) Incorrect. Since both statements are accurate, 'neither' is wrong.
Key point
Factoring 221 = 13 × 17 gives m=17 boys and n=13 girls (since m>n), confirming BOTH the total of 30 students AND the 78 girl-vs-girl games — both statements are correct.
Remember
221 boy-girl games → m=17 boys, n=13 girls → 30 total students, and 78 girl-girl games — both statements correct.
Common mistakes
- Not recognising that boy-girl game count = m × n, the key formula for this puzzle.
- Failing to factor 221 correctly into its prime components (13 and 17).
- Forgetting to apply the combination formula C(n,2) for counting girl-girl games.
Exam tip
Mixed-pair game count = m × n (boys × girls). Same-gender game count = C(m,2) or C(n,2). Two distinct combinatorics formulas for this classic puzzle type.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)