Could anyone please solve this problem? I guess, we can first select the president and vice and then the other 3; or we can select the first 3 members and then the president and vice. My question is should I add both of the counts? Or it would be only one of them?

Do you mean first three members?

yes, it was a mistake. I have corrected it.

Only one of them, since we have already taking the positions into consideration during our calculations, we need not arrange them once more

For first way you get:

12 * 11 * \frac{10!}{3! * 7!} = \frac{12!}{3! * 7!}

The other way you will get:

\frac{12!}{3! * 9!} * 9 * 8 = \frac{12! * 9 * 8}{3! * 9 * 8 * 7!} = \frac{12!}{3! * 7!}