A graduate student is looking for 5 members for his disserta

A graduate student is looking for 5 members for his dissertation committee. He has been working with 3 professors in math & 5 professors in physics. The math department demands that the chair of the dissertation committee be a mathematician. How many ways can the graduate student choose among the professors if the chair of the committee must be a mathematician and the rest of the committee can be a mixture of mathematicians and physicists?

Solution

There are 3 math and 5 physics professors

There must be atleast 1 mathematician in the 5 members

So mathematician and physician can be in any one of the following ways.

(1,4) (2,3) (3,2) (4,1) (5,0)

Total no of ways of selectng 5 from 8 in any way = 8C5 = 32

No of ways of selecting (1,4) (2,3) (3,2) (4,1) (5,0) = total no of ways - (0,5)

= 32-5c5 = 32-1 = 31 ways.