Choose groups by category of person Sorry for the vague title, I'm not really sure what the correct terminology for this type of questions is.
I have to hire 6 people: 2 programmers, 2 footballers and 2 logicians. I receive 90 applications for the positions. The number of people applying for each position is 


*

*25 footballers

*25 programmers

*20 logicians

*10 footballers who are also logicians

*9 programmers who are also footballers

*1 logician who is also a programmer


How many ways can I choose 6 applicants, given that some applicants are suitable for 2 positions?
So the number of applications for the footballer position is 44 (25 + 10 + 9). Therefore, if we ignore the other positions, the combinations of footballers I can choose is 946 (44 choose 2).  
However this ignores the other positions which may take away the footballers who are also programmers/logicians. Can anyone give me a hint on how to solve this?
Thanks
JP
 A: You're correct that the "naive" way to choose the 2 footballers would be $\begin{pmatrix} 44 \\ 2\end{pmatrix}$. Likewise, there are $\begin{pmatrix} 35\\2 \end{pmatrix}$ naive ways to choose the 2 programmers, and $\begin{pmatrix} 31\\2 \end{pmatrix}$ naive ways to choose the logicians, so that their product is the answer of "naive" ways to choose the six hires.
It's naive because we've certainly (as you point out) double counted some of our people, possibly hiring someone as both a footballer and logician for example. So, we've overcounted by some currently unknown number and have to revise our "naive" approach.
What scenarios have been overcounted? Hiring someone as a footballer and logician, for one. Second, hiring someone as both a footballer and programmer. Finally, we could double hire the programmer logician.
We might subtract off each of these counts from our naive answer, but does that take care of it? I think not; I think there are many many "bad" scenarios that this doesn't take care of.
We can "double hire" 0,1, or 2 footballers. Likewise with programmers, but we can only "double hire" 0 or 1 logicians. Thus, I think there are $3\cdot 3 \cdot 2 - 1$ cases that we need to subtract off: double hiring only a footballer; only 2 footballers; a programmer; 2 programmers; the logician; a single footballer and a single programmer; and so on. 
This sounds really computationally horrendous. I hope there's a simpler solution and someone else chimes in!
