Forming a committee

Question

Suppose a committee must be formed from a group of 15 professors and 10 administrators. How many committees can be formed if the committee must consist of 5 professors and 5 administrators?

Update 1: Suppose as above that we have 15 professors and 10 administrators to choose from. Now suppose that we must compose a committee of 4 which must either consist entirely of professors, or entirely of administrators. How much such committees can be formed?

The answer of Update 1 should be $$\binom{15}{4} +\binom{10}{4}$$ There should be summation because it says Either Or.

Correct me if I'am wrong?

Answer 1

We choose $5$ professors out of the $15$, and $5$ administrators out of the $10$. The answer is hence

$$\binom {15}{5}\times\binom{10}{5}$$

Answer 2

We choose $5$ committee members from 15 professors, and $5$ committee members from the 10 administrators.

That gives us $$\binom{15}5 \cdot \binom{10}5$$

possible ways of constructing a committee.