Question

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?

1. 39
2. 30
3. 21
4. 20

Answer 1

Correct Answer - Option 3 : 21

Let the number of males be M and the number of females be F.
Each male or each female will shake hands with each of the member of the opposite sex.
The total number of handshakes = M x F = 24
Possibilities for the number of males and females in the society (not necessarily in the same order) = (2, 12), (3, 8), (4, 6), (1, 24)
In case of (2, 12), the number of hugs would be = ²C₂ + ¹²C₂ =    \( 2!/(2! х 1) + 12!/(2! х 10!)\) = 1 + 66 = 67
Clearly, it is not in the option.
In case of (3, 8), the number of hugs would be = ³C₂ + ⁸C₂ = \( 3!/(2! х 1) + 8!/(2! х 6!)\) = 3 + 28 = 31
Clearly, it is not in the option either.
In case of (4, 6), the number of hugs would be = ⁴C₂ + ⁶C₂ = \( 4!/(2! х 2!) + 6!/(2! х 4!)\) = 6 + 15 = 21
Clearly, it is one of the options.

Hence, 21 possible hugs are there according to the given data.