<s>[INST] In how many ways can $5$ teaching posts be filled if $2$ posts are reserved for American candidates? 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.
 [/INST] 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
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