Slot Method vs Combinatorics

[happyface101]

Hello!

I have always wondered why the slot method is wrong here...

For example, if a committee has three spots, and there are 4 professors and 6 students being considered for the committee. How many ways can you construct the committee such that 1 professor and 2 students are on it.

My method -- wrong method -- to find ways the committee is filled by only 1 professor:
There are three slots, 4 professors, 6 students --> there are four ways that the first slot is filled by a professor, 6 ways the second slot is filled by a student, 5 ways the third slot is filled by a student (one student is already selected for the second slot) --> 4*6*5 = 120 ways the committee is filled by 1 professor and 2 students.

The correct way to find ways committee is filled by 1 professor:
4 choose 1 * 6 choose 2 -> (4! / (3!*1!)) * (6!/(4!*2!)) = 4 * 3 *5 = 60 ways

Can someone please tell me why the slot method is wrong? Thanks!

[tim]

The slot method, as I presume you are applying it, is only relevant for situations in which order matters. It doesn't matter which student is chosen first or second, so if you use the slot method, you have to divide your final answer by 2! in order to account for the two different ways the same pair of students could be chosen if they were selected sequentially. If you understand the combinatorics approach, I recommend using that as your method for all problems of this type, as it tends to bypass altogether the question of whether order matters.