There are at least eight versions of this problem.

1. Order does not matter at all: (1,2,3,4,5,345) and (345,1,3,5,2,4) are the same.
   
   
2. Cyclic order matters: (1,2,3,4,5,345) = (2,3,4,5,345,1) but not (1,3,5,2,4,345) [this and the next are rather natural for angles addign to 360 degrees!)
   
   
3. Cyclic order matters, reflections identified: (1,2,3,4,5,345) = (1,345,5,4,3,2) but not (1,3,5,2,4,345)
   
   
4. All order matters.

1. Zero angles forbidden
   
   
2. Zero angles permitted

4. b) is the easiest. It's solved most simply by thinking of the problem as sticking five separators into a deck of 360 cards - and the number is hence just 365 choose 5 . That's 365*364*363*362*361/5*4*3*2*1 .
   
   

4. a) - to avoid empty sets, remove 6 cards, place the separators ina 354-card deck, and put the six back. 359 choose 5 ways

The others are more complicated, and would be computed using symmetries and "Polya's method." On the principle that most partitions of a big number intoa small number of parts have no repetitions, they can be approximated fairly well by dividing (4a) and (4b) by 720, 6, and 12 respectively.

Good Hunting!
RD