In how many ways can the officers of the student council be selected?

Question

This is the question,

A student council has $10$ members. From this one President, one Vice-President, one Secretary, one Joint-Secretary and two Executive Committee members have to be elected. In how many ways this can be done?

1. $151200$
2. $75600$
3. $37800$
4. $18900$

And this is the answer on almost every website

Given, A student council has $10$ members. From this one President, one Vice-President, one Secretary, one Joint, one Secretary and two Executive Committee members have to be elected.

$\implies 6$ members are elected out of $10$ members. The number of ways to elect $6$ members out of $10$ members = $P(10,6) = 151200$.

This seems wrong to me. I think the answer should be like,

Out of $10$ we first select $1$ president, then out of $9$ one vice president, out of $8$ one Secretary, out of $7$ one joint Secretary and then out of $6$ two Executive Committee members.

This give us $\binom {10}{1} \times \binom {9}{1} \times \binom {8}{1} \times \binom {7}{1} \times \binom {6}{2} = 75600$.

Am I wrong somewhere?

Answer 1

Since an answer affirming the book answer has been put, I disagree with it.

Nowhere is it mentioned that particular seats are associated with the two Executive Committee members, your answer is correct, let us confirm it in a skightly different way, first choose the six who will be on the council, and then select people with "solo" designations, the remaining two will just be members

$\binom{10}{6}\cdot6*5*4*3 = 75600$