The prime factorization of 2160 is (24)(33)(5). Any divisor of 2160 is a combination of one or more of these 8 factors.
Order never matters in multiplication because multiplication is commutative. That means you need to think of combinations, rather than permutations. But you have to think about duplications. For example, if you take all the threes, the five and the first two twos and multiply those, you get the same product as if you had taken the second pair of twos and all the threes and the five.
A good way to approach this problem is to split it up: you can take from 0 to 4 of the twos and from 0 to 3 of the threes and from 0 to 1 of the fives.
You can solve part b by subtracting the number of odd divisors from your answer to part a. The odd divisors you get by taking 0 of the twos and from 0 to 3 of the threes and from 0 to 1 of the fives.
Hope this helps,
Stephen and Penny