49 members attended a party. Among them, 22 are males and 27 are females.…
2025
49 members attended a party. Among them, 22 are males and 27 are females. Handshakes took place between males, between females, and between males and females — but only 12 people took part, with each of them shaking hands with every other one exactly once. How many handshakes took place in total?
- A.
66
- B.
78
- C.
55
- D.
91
Attempted by 1 students.
Show answer & explanation
Correct answer: A
A handshake is an unordered pairing of two people, so the number of possible handshakes among n people equals the number of ways to choose 2 people from n, given by the combination formula C(n,2) = n(n-1)/2. This depends only on how many people actually take part, not on any other grouping among them.
Of the 49 people at the party, only 12 actually shook hands with anyone; the remaining people and the male/female breakdown (22 and 27) describe the party but not who took part in the handshaking.
So the handshaking population is exactly these 12 people, and every handshake is one unordered pair chosen from this group of 12.
Apply the combination formula with n = 12: C(12,2) = 12 × 11 / 2 = 66.
Cross-check: this can be verified by summing the handshakes contributed as each person joins in turn: the first person shakes hands with the other 11, the next new person shakes hands with the remaining 10, and so on, giving 11 + 10 + 9 + ... + 1 = 66 — the same total.
So 66 handshakes took place among the 12 people who participated.