In how many ways can a committee consisting of $1$ president, $1$ vice-president and three members from a poll of $12$ people be selected? We have to form a committee consists of $1$ president, $1$ vice-president and $3$ members from a poll of $12$ people. How many ways can we do it? Is not this simply a selecting $5$ people from $12$ $$\frac{12!}{5!7!}$$ type of problem? If not, then can you please explain why? Thanks in advance!

No, it is not. After having chosen $5$ people, you still have to decide which one of them will be president and wich one will be vice-president. Therefore, you'all have to multiply your answer, which is $\binom{12}5$, by $20(=5\times4)$.