The correct answer to this problem is given by the binomial coefficient

which reaches a maximum of 252 for *k* = 5. The number of committees of *k* members equals the number of committees of (10 - *k*) members, because any committee of *k* members defines a unique group of (10 - *k*) non-members.

Answering this question without computation involves mentally constructing committees of *k* members and evaluating their number by the ease with which they come to mind. Committees of few members such as 2, are easier than committees of many members, such as 8. The simplest scheme for the construction of committees is a partition of the group into disjoint sets. It is easy to construct five disjoint committees of 2 members, but impossible to generate even two disjoint committees of 8 members. So, using imaginability alone, the small committees will seem far more numerous than larger committees, whereas in fact there is a bell-shaped function.