Question

The local sci club has 100 members. At a meeting, a public vote shows that 43 members are fans of Arthur C. Clarke and 41 members are fans of Robert A. Heinlein. The random selection of a board of directors of at least how many members will guarantee that there are at least 5 Clarke fans, and also at least 5 Heinlein fans among the directors?

If n is the right answer, you must not only explain why the random selection of n members suffices, but also why selecting fewer than n members may not suffice.

Answer 1

The minimum number of board members required to guarantee at least 5 Clarke fans and 5 Heinlein fans is 13. Selecting fewer than 13 members may not suffice because it's possible to have fewer than 5 fans of either author. This calculation uses the Pigeonhole Principle and considers the maximum number of fans before guaranteeing the requirement.

Step-by-step explanation:

To determine the minimum number of board members needed to guarantee at least 5 fans of Arthur C. Clarke and also at least 5 fans of Robert A. Heinlein, we must take into account the worst-case scenario. This is a problem of combinatorics where we use the principle of the Pigeonhole Principle. First, let's consider selecting the maximum number of members that do not meet either requirement. We could select 4 Clarke fans and 4 Heinlein fans for a total of 8 members without satisfying either condition. If we add one more person to this group, regardless of their preference, we cannot avoid having at least 5 fans of one author. Therefore, to ensure we do not meet the condition, we cannot select more than 8 members.

To ensure we have at least 5 fans of each, we need to add the next possible number of members who are fans of neither Clarke nor Heinlein. Since we have 100 members and only 84 (43+41) have been accounted for as fans of either, there are at least 16 members who could be fans of neither. Adding 4 of these to the selected 8 (to keep the total number of fans below 5 for each) would give us 12 members. Now, if we randomly select one more member to this group of 12, we are sure to have at least 5 fans of either Clarke or Heinlein, and possibly both, because we would be selecting from those who are fans. Thus, the minimum board size to guarantee at least 5 fans of each author is 13 (8 for Clarke and Heinlein fans and 4 from the possible neutral members plus one additional member).

The reason why selecting fewer than 13 members may not suffice is because with 12 or fewer members, there is no guarantee to meet both conditions of having 5 Clarke fans and 5 Heinlein fans. The selection could be such that you end up with fewer than 5 fans of either author.