What does this count?

Question

What does this count? ${w\choose3} \cdot {m\choose2} + {w\choose4} \cdot m + {w\choose5} $

(a) The number of ways to choose $5$ people out of a group consisting of $w$ women and $m$ men, where at most $3$ women can be chosen.

(b)The number of ways to choose $5$ people out of a group consisting of $w$ women and $m$ men, where at most $3$ men can be chosen.

(c) The number of ways to choose $5$ people out of a group consisting of $w$ women and $m$ men, where at least $3$ women must be chosen.

(d) The number of ways to choose $5$ people out of a group consisting of $w$ women and $m$ men, where at least $3$ men must be chosen

Not too sure why its (c), can someone explain? I thought it was (b) because of the ${m\choose2}$ and $m$ parts. Why is it only $5$ people as well?

Answer 1

And means product, here are all the case:

• $3$ Women and $2$ Men: $\displaystyle {w \choose 3}{m \choose 2}$

• $4$ Women and $1$ Man: $\displaystyle {w \choose 4}{m \choose 1}$

• $5$ Women: $\displaystyle {w \choose 5}{m\choose 0}$

where $\displaystyle {m \choose 1}=m, {m\choose 0}=1$.

Thus the result is: $\displaystyle {w \choose 3}{m \choose 2}+{w \choose 4}m+{w \choose 5}$ and this is at most $2$ Men as you can see.

In other words:

At most $2$ men $=$ at least $3$ Women (Choice $c$) [The Correct Answer]

At most $3$ men $=$ at least $2$ Women (Choice $b$)