Discussion
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 hand shakers. Which one among the following numbers indicates the possible number of hugs?
Answer: C
Let there be \( m \) men and \( f \) women. Handshakes happen only between opposite sexes and hugs only within a sex.
\[ mf = 24, \qquad \text{hugs} = {}^{m}C_{2} + {}^{f}C_{2} \]
Testing the factor pairs of 24:
- \( 2 \times 12 \): \( 1 + 66 = 67 \)
- \( 3 \times 8 \): \( 3 + 28 = 31 \)
- \( 4 \times 6 \): \( 6 + 15 = 21 \)
So 21 is attainable, with four of one sex and six of the other.
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