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?

Let M be the number of males and F be the number of females in the group. 1. Handshakes: Handshakes occur between friends of opposite sex. The total number of handshakes is M * F. Given, M * F = 24. 2. Hugs: Hugs occur between friends of the same sex. Number of hugs among males = M * (M - 1) / 2 Number of hugs among females = F * (F - 1) / 2 Total number of hugs = [M * (M - 1) / 2] + [F * (F - 1) / 2] 3. Find possible integer pairs for (M, F) such that M * F = 24: The possible pairs are (1, 24), (2, 12), (3, 8), (4, 6), and their reverses (24, 1), (12, 2), (8, 3), (6, 4). Since the formula for hugs is symmetric with respect to M and F, we only need to calculate for one order of each pair. a) If (M, F) = (1, 24): Hugs = [1 * (1 - 1) / 2] + [24 * (24 - 1) / 2] = 0 + (24 * 23 / 2) = 0 + 12 * 23 = 276 b) If (M, F) = (2, 12): Hugs = [2 * (2 - 1) / 2] + [12 * (12 - 1) / 2] = (2 * 1 / 2) + (12 * 11 / 2) = 1 + 6 * 11 = 1 + 66 = 67 c) If (M, F) = (3, 8): Hugs = [3 * (3 - 1) / 2] + [8 * (8 - 1) / 2] = (3 * 2 / 2) + (8 * 7 / 2) = 3 + 4 * 7 = 3 + 28 = 31 d) If (M, F) = (4, 6): Hugs = [4 * (4 - 1) / 2] + [6 * (6 - 1) / 2] = (4 * 3 / 2) + (6 * 5 / 2) = 2 * 3 + 3 * 5 = 6 + 15 = 21 4. Compare with options: The possible numbers of hugs are 276, 67, 31, 21. Option C is 21, which is one of the possible numbers of hugs. The final answer is C) 21.