Final Exam, Wed 28 March 2001, 8-12 a.m.

- A mid-size university department has six working groups
developing the department's Teaching Quality Initiatives.
Five professors (A,...,E) participate in these groups,
according to the following scheme:
1: A, B, C 4: B, C, D 2: A, C, D 5: B, E 3: A, E 6: C, D, E - The student association of the abovementioned department
selects ten representatives to work on the Teaching Quality
Initiatives, so that each of these ten persons participates
in the workings of exactly one of the groups 1-6. Compute
the number of ways the groups' student representations can
be arranged, under the conditions that each group must
have at least one student member, but at most as many as there
are professor members in that group.
(For the purposes of this problem, students are considered
to be identical, so that it is only significant how
*many*student representatives there are in each working group.) - Our example department has been awarded for its development work
a Quality Diplom by the Rector, and each of the professors
A,...,E may now hold the Diplom on their bookshelf one
week at a time. Design for the Diplom a 10-week rotation
schedule, during which each pair of professors who had worked
together in some of the groups 1-6 exhanges the diplom
exactly once. (The direction of the exchange, i.e. who
forwards the Diplom to whom, does not matter.)
- Find a closed form solution for the recurrence equation:

- A round pizza is cut into pieces by knife cuts,
so that any two cuts meet pairwise,
but no three at the same point (cf. Figure 1).
Prove that the cuts divide the pizza into
pieces.
(
*Hint:*One way to solve the problem is to view the pizza as a plane graph.)

*Grading:* Each problem 12 points, total 60 points.