Picking at least 2 specific members for committee?

Question

I have a club with 20 people including 5 directors, and I need to form a committee with 5 people with at least 2 directors, how many possible ways is there to do this?

I see why the outcomes can't be: ${5 \choose 2} * {18 \choose 3}$

because it will over count some possibilities.

Would the possibilities be something like this? ${5 \choose 2} * {15 \choose 3} + {5 \choose 3} * {15 \choose 2}+{5 \choose 3} * {15 \choose 2} + {15 \choose 1}*{5 \choose 4}+{5 \choose 5}$

Answer 1

You almost got it right. The answer is this

${5 \choose 2} * {15 \choose 3} +{5 \choose 3} * {15 \choose 2} + {5 \choose 4}*{15 \choose 1}+{5 \choose 5}$

ie You have 2 ,3 ,4 or 5 directors

Answer 2

It's $\binom52 \binom{15}3+\binom53\binom{15}2+\binom54 \binom{15}1+\binom55\binom{15}0$, which is what you have.