A three-person committee must be chosen from a group of

Gmat_mission wrote:A three-person committee must be chosen from a group of 7 professors and 10 graduate students. If at least one of the people on the committee must be a professor, how many different groups of people could be chosen for the committee?

A. 70
B. 560
C. 630
D. 1,260
E. 1,980

[spoiler]OA=B[/spoiler].

Could anyone give me some help here? Please. I would like to know how to solve this PS question. <i class="em em-sob"></i>

Hi gmat_mission.

Here is how I'd solve it.

A three-person committee ------------> this implies that we have to pick a total of 3 people.
7 professors -------------------------------> Group 1 of people.
10 graduate students --------------------> Group 2 of people.

If at least one of the people on the committee must be a professor --------> this implies that we could pick 1 professor, 2 professors or 3 professors.

How many different groups of people could be chosen for the committee?

Now, the number of ways of forming the committee is:

- If we pick only one professor and two students:
$$7\ C\ 1\cdot10\ C\ 2\ =\ \frac{7!}{6!}\cdot\frac{10!}{8!2!}=7\cdot45=315.$$

- If we pick two professors and one student:
$$7\ C\ 2\cdot10\ C\ 1\ =\ \frac{7!}{5!2!}\cdot\frac{10!}{9!}=21\cdot10=210.$$

- If we pick three professors and none students:
$$7\ C\ 3\cdot10\ C\ 0\ =\ \frac{7!}{4!3!}\cdot1=35.$$

Finally, we have to add the results, hence we get: 315+210+35=560.

Therefore, the correct answer is the option B.

I hope it helps.