Here's my take:
A baking club wants to form an executive committee. There are $18$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $2$ people?
It's just \(\binom{18}{2}\)because you just have to choose \(2\) people out of the 18 candidates.
\(\binom{18}{2}\)= \(\boxed{153}\).
A baking club wants to form an executive committee. There are $18$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $2$ people, including Mark?
Well, one of the spots is taken (by Mark), and there are 17 candidates left, with 1 spot left, so it's just \(\binom{17}{1}\), or \(\boxed{17}\).
A baking club wants to form an executive committee. There are $18$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $2$ people, not including Mark?
For this one, because Mark is no longer a candidate that can be put on the committee, that would leave 17 candidates, and 2 need to be chosen, so it would just be \(\binom{17}{2}\), or \(\boxed{136}\)
What do you notice about your answers to all three parts?
The answers of the second and third parts add up to the answer of the first part. This makes perfect sense because the committee either has Mark on it, or it doesn't.
Hope this helps!
