There are at least eight versions of this problem.
- Order does not matter at all: (1,2,3,4,5,345) and (345,1,3,5,2,4) are the same.
- 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!)
- 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)
- All order matters.
- Zero angles forbidden
- Zero angles permitted
- 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 .
- 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.