A project-based course requires that students be assigned to project teams at the start of the term. For this purpose, each student is asked to examine the set of projects available and to indicate a first, second, and third preference for a project assignment. When these preferences are collected, the instructor assigns the students to project teams, aiming for an optimal assignment of students to teams. This year, there are ten available projects and 16 students enrolled. There is a maximum team size, either 2 or 3, on each project, according to the nature of the work to be done. It is permissible for a student to work alone on a project (that is, in a team of size 1). A student may not be assigned to more than one project. The table below shows the student preferences and the maximum team sizes. A preference number of 3 represents the strongest preference.

S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12	S13	S14	S15	S16	limit
P1	2	0	1	0	0	0	3	0	0	2	0	0	0	2	3	0	3
P2	0	0	0	2	2	2	0	0	0	0	2	0	0	0	0	3	3
P3	3	3	0	3	3	3	0	0	0	0	3	0	0	0	0	0	3
P4	0	2	2	0	0	0	0	3	3	0	0	0	3	3	0	0	3
P5	1	0	0	0	0	1	0	0	0	0	0	2	0	0	0	0	2
P6	0	1	0	1	0	0	2	2	2	3	1	3	2	0	2	0	2
P7	0	0	3	0	1	0	1	1	0	0	0	0	0	1	1	1	2
P8	0	0	0	0	0	0	0	0	1	1	0	1	1	0	0	2	2
P9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2
P10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2

1.) Suppose we wish to maximize the sum of the preferences for the assignments made in the class. The ideal value would be 48, if every student were assigned to their preferred project, but that may not be possible. It is not necessary for all projects to be completed. What is the largest possible value and what is the assignment that produces it?

2.) Suppose, unlike (a) we want to make sure all projects are completed. How does this change the preference value and what is the assignment that produces it?

3.) Building from scenario (b), suppose students 4 and 11 have a medical condition that does not allow them to work on project P3 (even though they put it as a preference). What is the preference value and the assignment that produces it?