TITLE: Choosing office hours to maximise number of students who can attend at least one time slot
QUESTION [49 upvotes]: I have the following (real world!) problem which is most easily described using an example.
I ask my six students when the best time to hold office hours would be. I give them four options, and say that I'll hold two hours in total. The poll's results are as follows (1 means yes, 0 means no):
$$\begin{array}{l|c|c|c|c} 
\text{Name} & 9 \text{ am} & 10 \text{ am} & 11 \text{ am} & 12 \text{ pm} \\ \hline
\text{Alice} & 1 & 1 & 0 & 0 \\ 
\text{Bob} & 1 & 1 & 0 & 1 \\ 
\text{Charlotte} & 1 & 1 & 0 & 1 \\ 
\text{Daniel} & 0 & 1 & 1 & 1 \\
\text{Eve} & 0 & 0 & 1 & 1 \\
\text{Frank} & 0 & 0 & 1 & 0 \\ \hline
\text{Total} & 3 & 4 & 3 & 4
\end{array}$$
Naively, one might pick columns 2 and 4, which have the greatest totals. However, the solution which allows the most distinct people to attend is to pick columns 2 and 3.
In practice, however, I have ~30 possible timeslots and over 100 students, and want to pick, say, five different timeslots for office hours. How do I pick the timeslots which maximise the number of distinct students who can attend?

REPLY [3 votes]: Practically speaking, simply vary the office hours each week.