1. I liked this one-- 
  Part a just took some playing with examples and it was easy enough that
	it's okay that part b depends on it
  The structural induction part was quite easy--I'd just put it off till I 
	worked on some of the others.  

2. I gave up on this one--I got the initial equation right but couldn't change 
	it into a closed form.  Skipped part b since I didn't get part a.

3.  I see you've added another part since I took it, but the (a) was
	easy if you remember Fermat's theorem.
    (b) seemed familiar--was this a recitation problem.  easy enough.

4.  (a) was easy
    I set (b) aside for a while then saw it was stars and bars and it was
	easy then.
    (c) fine.
    (d) recognized it was inclusion-exclusion but I wasn't careful with 
	those terms--forgot the last half of the paths after the pits.
    (e) comb proof pretty straightforward.  and easy to verify.

5.  good simple problems for probability and using previous parts.
	i just had arithmetic mistakes that threw me off.
	maybe don't need 2 independence problems