In a group of children, each child exchanges a gift with every other child. If the number of gifts is 132, then the number of children in the group is
(a)10
(b)11
(c)12
(d)13
Answer
Answer (as printed): C
Explanation
Let there be $n$ children in the group. Then, first child exchanges gifts with other $(n-1)$ children; second child exchanges gifts with other $(n-2)$ children; and so on. So, number of gifts $=2[1+2+\ldots . .+(n-1)]$ $$=2 \times \frac{n(n-1)}{2}=n(n-1) . \begin{aligned} \therefore \quad n(n-1)=132 & \Leftrightarrow n^{2}-n-132=0 \\ & \Leftrightarrow n^{2}-12 n+11 n-132=0 \\ & \Leftrightarrow(n-12)(n+11)=0 \Leftrightarrow n=12 . \end{aligned}$$
Explanation as extracted from the printed page; notation may be imperfect.