Counting without Counting

Let me tell you a little dirty secret about the mutual dislike between mathematicians and physicists.  It can escalate into a full blown war if diplomacy is not attempted properly.

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.

Now comes the monkey wrench.

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.  Yes, they are just a bunch of you know what (click on the link ) in the mathematics department!

For comic relief, can you figure out how may 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?