50 States w/2 senators each. Probablity of various committee compositions From Carol Ash's, "Probability Tutoring Book".  pg. 28 section 1-4, question 10.

There are 50 states and 2 senators from each state.  Find the prob
  that a committee of 15 senators contains:
a. at least 1 from each of Hawaii, Mass. and Penn.
b. at least 1 from the 3 state region composed of Hawaii, Mass. and
  Penn.
c. 1 from Hawaii and at least 1 from Mass.

Is my answer below correct?
 A: Your answers all overcount, I’m afraid. For (a) consider a committee that contains both senators from Hawaii and one senator each from Massachusetts and Pennsylvania. If the senators from Hawaii are A and B, you count this committee once with A as the Hawaiian senator counted in the $2^3$ factor and B as one of the $12$ counted in the $\binom{97}{12}$ factor, and once with B as the Hawaiian senator counted in the $2^3$ factor and A as one of the $12$ counted in the $\binom{97}{12}$ factor. Every committee that contains two senators from at least one of the three named states is overcounted. 
The other calculations overcount in similar fashion.
A: She answered all these questions using inclusion-exclusion, however I answered  with direct calculation.  Are my answers below correct?  If not, why?
My thought is that when we say "exactly" or "at least" I pick a member of a group to meet the condition.  If it says exactly I eliminate the remaining members of that group before picking the remainder, or if it says at least I leave them in and then pick the remainder.
a. $\huge\frac{2^3\binom{97}{12}}{\binom{100}{15}}$
b. $\huge\frac{6\binom{99}{14}}{\binom{100}{15}}$
c. $\huge\frac{2^2\binom{97}{13}}{\binom{100}{15}}$
