a)2m 2 ways is correct
b)2,4,6 is correct as well
c)Counting these would be very hard especially with a length of 6. So one much do it mathmatically. The ant's first move has 3 choices. His next move has 2 choices, and same with all the rest of his moves, until the last one at which time he has only one move which is to end at b.
6 moves have these many choices:
There are 3 choices times 2 paths times 2 paths times 2 paths times 2 paths and ending at b which there is only one path left.
(im not 100% sure on this but i am fairly certain)
d)I cant do it...There might be some way. Sorry.
EDIT: I doubt there is a way for d to work ive tried it alot.