Handshakes at a party — Aptitude Question
Handshakes at a party: At a party every person shakes hands exactly once with every other person, making 66… Answer and worked solution on CodeKairo.
- Topic: Permutations & Combinations
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 50 seconds
Question
At a party every person shakes hands exactly once with every other person, making 66 handshakes in all. How many people are there?
Options
- 11
- 12
- 13
- 22
Show the answer and the worked solution
Correct answer: B. 12
Worked solution
- Handshakes = C(n,2) = n(n − 1)/2 = 66, so n(n − 1) = 132.
- 12 × 11 = 132, so n = 12.
Answer: 12.
The idea
Handshakes, matches in a league, diagonals plus sides of a polygon — all are C(n,2). Solve n(n − 1) = 2k by finding consecutive factors.
Shortcut. n(n − 1) = 132 and √132 ≈ 11.5, so n and n − 1 straddle it: 12 × 11 = 132, giving n = 12.
Common trap. 11 — picking the smaller factor of 12 × 11; 11 people make only 11 × 10 ÷ 2 = 55 handshakes.
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