UPSC Prelims 2010 Q115: Science & Technology – Quantitative Aptitude
When ten persons shake hands with one another, in how many ways is it possible?
- (a) 20
- (b) 25
- (c) 40
- (d) 45
Answer: (d) 45
Explanation
Counting ways ten persons shake hands with one another
When ten persons each shake hands with every other person exactly once, each handshake involves a unique pair — a classic combinations problem.
| Step | Calculation |
|---|---|
| Formula | n(n−1)/2, where n = number of persons |
| Substitute n = 10 | 10 × 9 / 2 |
| Compute | 90 / 2 = 45 |
Why this formula applies:
| Point | Detail |
|---|---|
| Each person shakes hands with | 9 other persons |
| Total person-to-person handshakes | 10 × 9 = 90 (double-counted, once per person in the pair) |
| Unique ways to shake hands | Divide by 2: 45 |
Answer: (d) 45.
Why each option is right or wrong
- (a) Incorrect. 20 drastically undercounts the actual number of ways ten persons can shake hands with one another.
- (b) Incorrect. 25 also undercounts the correct total.
- (c) Incorrect. 40 is close but does not match the exact formula result.
- (d) CORRECT. Using n(n−1)/2 with n=10 gives 10×9/2 = 45 ways the ten persons can shake hands with one another.
Key point
When ten persons each shake hands with every other person exactly once, the total number of ways = n(n−1)/2. With n=10, that's 10×9/2 = 45.
Remember
10 people, handshakes with everyone once: 10×9/2 = 45.
Common mistakes
- Forgetting to divide by 2, double-counting each handshake from both participants' perspectives.
- Using an incorrect value of n or miscounting the 'others' each person interacts with.
- Confusing this with a different combinatorial setup, like ordered pairs instead of unordered ones.
Exam tip
Handshake/pairwise-comparison formula: n(n−1)/2 — the same combinatorics logic as round-robin tournaments and pairwise rankings.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)