There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the following questions based on this information.
(a) How many ways can the committee be formed from the supervisors available?
(b) How many ways can the committee be formed if the General Secretary must be a female?
(c) How many ways can the committee be formed if it must have at least one man and at least one woman?
(a) $\binom{17}{3}$
(b) $\binom{16}{2}$
(c) $\binom{10}{1}\binom{7}{2} + \binom{10}{2} \binom{7}{1}$.
Do you think my answers are correct?
Best Answer
Your answers look mostly correct. There is an easier way to do $(c)$ though. Start by picking a man and a woman. That gives us $\binom{10}{1} \cdot\binom{7}{1} = 70$. We then choose one person out of the $15$ remaining. We then have to permute the members, so we multiply by $3!$. So our answer is $70 \cdot15 \cdot3!$. However, we also have to divide out by $2$ to handle the symmetry cases. So if $m_{1}$ is drawn first and $m_{2}$ is drawn second, the permutations handle the ordering of those.
Note- for all of your answers, since the committees have labeled positions, you have to permute the members through the positions. So you have to multiply by $3!$ for each of your solutions.