A committee of $8$ workers is formed selecting from a group of $6$ men and $5$ women. How many different committees can be formed if the committee…

Question

I need help with a question. I have absolutely no idea how to do this, can someone please explain how to solve this problem:

A committee of $8$ workers is formed selecting from a group of $6$ men and $5$ women. How many different committees can be formed if the committee should contain exactly $5$ men?

Answer 1

Hint: choose the $5$ men out of $6$ and note that you have to choose $3$ women out of $5$

Answer 2

Well select $5$ of $6$ men and then $3$ out of the $5$ women. So $$\binom{6}{5}\binom{5}{3}$$

Answer 3

We can deal with the two seperately.

For men, we choose 5 out of 6, so

$ {6}\choose{5} $ = $ {6}\choose{6-5} $ = $ {6}\choose{1} $ = $6$

Now for women, we choose 3 out of 5, so

$ {5}\choose{3} $ = $ \frac{5*4*3}{3*2*1} $ = $\frac{60}{6} = 10$

Since the two are independent events, we multiply them together, $6*10=60$