Probability of 3 friends being part of 5 person committee

Question

There are $18$ students that will be forming a committee of $5$ people. The people in the committee will be picked out of a hat randomly. What is the probability that Joe, Joseph, and John will be $3/5$ members of the committee?

I've tried during various things, I know that the total number of combinations of 5 people that can be formed from $18$ is $8568$ $(18C5)$, but I'm not exactly sure what the numerator should be. I would love an explanation as to why the answer is why it is please

Answer 1

You've got one part of the answer: all of the possible committees which is $_{18}C_5$.

Now you'll need the number of committees that have Joe, Joseph, and John.

As a hint, you know three of the five committee members -- Joe, Joseph, and John -- so you only need to choose two from who's left.

Can you take it from here?

After picking Joe, Joseph, and John, there are $15$ left, and we need to pick two, which is $_{15}C_2$. So the probability is $(_{15}C_2)/(_{18}C_5)$.