There are 6 persons arranged in a row. Another person has to shake hands with 3 of them so that he should not shake hands with two consecutive persons. In how many distinct possible combinations can the handshakes take place?
Answer & explanation
Answer: (b) 4
Choosing 3 of 6 people in a row with no two next to each other leaves only {1, 3, 5}, {1, 3, 6}, {1, 4, 6} and {2, 4, 6}. That is 4 combinations, matching C(4, 3).
- Number the persons 1 to 6 and choose 3 with no two adjacent.
- Possible sets: {1, 3, 5}, {1, 3, 6}, {1, 4, 6}, {2, 4, 6}.
- So there are 4 combinations.
- Check: C(n − k + 1, k) = C(6 − 3 + 1, 3) = C(4, 3) = 4.
Remember · Ways to pick k non-adjacent items from n in a row = C(n − k + 1, k).
Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·