I need help in trying to understand the answer to this exercise.
[Question]
A club is considering changing its bylaws. In an initial straw vote on the issue, 24 of the 40 members of the club favored the change and 16 did not. A committee of 6 is to be chosen from the 40 club members to devote further study to the issue.
How many of the committees will contain at least 3 club members who, in the preliminary survey, favored the change in the bylaws
[Answer]
$\binom{24}{3} \binom{16}{3} + \binom{24}{4}\binom{16}{2} + \binom{24}{5}\binom{16}{1} + \binom{24}{6}\binom{16}{0} = 3,223,220$
I don't completely understand the reasoning for building the equation in this way. I understand that 3 + 3 = 6 and that is why I have $\binom{24}{3} \binom{16}{3}$. But I don't understand why I have to use $\binom{16}{3}$ in the first place. Also, what is the purpose of applying both the multiplication rule and the addition rule? Could someone try to explain this please?
Thanks,
Tony
Think of it this way: how can you select a committee with exactly three of the special type? There's 24 to choose from, so there's $\binom{24}{3}$ ways to do that. You also have to pick three from the rest of the group and there are $\binom{16}{3}$ ways to do that, since 16 members did not favor the change. So there are $\binom{24}{3} \binom{16}{3}$ ways to choose a 6 person committee with exactly 3 of each type.
Similarly for 4,5, and 6 (since it's at least 3), so you have to sum them all up.