a) 3 of your letters are used up by choosing vowels. So you have 2 choices left out of 7: (10-3)C(5-3) = 7C2 = 21
b) You still need to select 5 letters, but your set is smaller (10-3)C5 = 7C5 = 21
c) This should be the set of all choices minus the set of no vowels. 10C5 - 7C5 = 252 - 21 = 231
I am moderately confident in my answers, but I haven't looked at these in a while, so take it with skepticism.
edit: I think I did a wrong, I calculated how many vowels instead of how many consonants.