In a society it is customary for friends of the same sex to hug and for friends of opposite sex to shake hands when they meet. A group of friends met in a party and there were 24 handshakes. Which one among the following numbers indicates the possible number of hugs ?
- (a) 39
- (b) 30
- (c) 21 ✓ UPSC's answer
- (d) 20
Why the answer is (c)
• Let $m$ be the number of men and $w$ be the number of women in the group.
• Handshakes occur only between opposite sexes, so the total number of handshakes is $m \times w = 24$.
• Hugs occur only between friends of the same sex, so the total number of hugs is $\binom{m}{2} + \binom{w}{2} = \frac{m(m-1)}{2} + \frac{w(w-1)}{2}$.
• We need to find integer pairs $(m, w)$ such that $mw = 24$. The possible factor pairs are $(1, 24), (2, 12), (3, 8), (4, 6)$ and their reverses.
• Calculating hugs for each pair: $(1, 24) \rightarrow 0 + 276 = 276$; $(2, 12) \rightarrow 1 + 66 = 67$; $(3, 8) \rightarrow 3 + 28 = 31$; $(4, 6) \rightarrow 6 + 15 = 21$.
• Among the calculated values, only 21 appears in the options, corresponding to the group composition of 4 men and 6 women (or vice versa).
Why the other options are wrong
- (a) 39
- 39 is not a possible number of hugs for any integer pair of men and women whose product is 24.
- (b) 30
- 30 is not a possible number of hugs for any integer pair of men and women whose product is 24.
- (d) 20
- 20 is not a possible number of hugs for any integer pair of men and women whose product is 24.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.