It happens that a graduate student in theoretical/mathematical physics is lookin

Question

It happens that a graduate student in theoretical/mathematical physics is looking for five members of his dissertation committee. He has been working closely with three professors in the mathematics department, and 5 professors in the physics department on his dissertation research.
The mathematics department demands that the chair of the dissertation committee must be a mathematician in order to keep the physicists in check. What? How arrogant! But, there is no other choice if a dissertation committee has to be assembled in time. As usual, physicists have to swallow their pride in order to keep peace.
For comic relief, can you figure out how many ways can this hapless graduate student can choose among his beloved professors if the chair of the committee must be a mathematician, and the rest of the committee can be a mix of mathematicians and physicists?

Explanation / Answer

three professors in the mathematics department, and 5 professors in the physics department are there

committee members = 5

Let the total set of possible people be: M1, M2, M3, P1, P2, P3, P4, P5

The chair must come from the Math Department: There are three choices.

The other four come from the remaining 7 (in any permutation; it’s a combination), so there are

      C(7,4) combinations

      C(n,r) = n! / (r! * (n-r)!)   =   n(n-1)(n-2). . . (n – r + 1) / r!

      C(7,4) = 7! / (4! * 3!)    = ( 7*6*5*4) / (4*3*2*1)   = 35

So, for each of the 3 choices for the chair, there are 35 choices for the rest of the committee. That gives 3*35 = 105 total combinations.

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