In how many ways can $5$ teaching posts be filled if $2$ posts are reserved for American candidates?

Question

For the post of $5$ teachers, there are $23$ applicants, $2$ posts are reserved for American candidates and there are $7$ American candidates among the applicants. In how many ways can the selection be made ?

Answer in textbook is given as

selection of $2$ American candidates out of $7$ + selection of $3$ from $16$ others left

$$\binom{7}{2} \times \binom{16}{3} $$

But, I am using combinations of $2$ Americans and $3$ others from left $16$ + combinations of $3$ Americans and $2$ others from left $16$ + combinations of $4$ Americans and $1$ other from left $16$ + combinations of $5$ Americans $$\binom{7}{2} \times\binom{16}{3} +\binom{7}{3} \times\binom{16}{2} +\binom{7}{4} \times\binom{16}{1}+\binom{7}{5} \times\binom{16}{0}$$

Is it wrong? Why? Please explain.

Answer 1

I think problem is here is interpretation of question by me. The book answer interprets the question as saying there must be exactly two Americans, whereas I have interpreted the question as saying there must be at least two Americans. It's a poorly stated question

Answer 2

this is because the no of ways you can select 2 American candidates out of 7 is 7C2-------M

and the number of ways you can select the rest is (23-7)C(5-2)-------------N

Thus, total ways = M*N = 7C2 * 16C3