Question

How many different 8 professionals committees can be formed each
containing at least 2 Professors, at least 2 Technical Managers and 3
Database Experts from list of 10 Professors, 8 Technical Managers
and 10 Database Experts?

Answer 1

The 8 members team can be either of the following:
a. 3 professor, 2 technical managers and 3 database expert
Total ways of choosing this =10C3*8C2*10C3=120*28*120=403200

b. 2 professor, 3 technical managers and 3 database expert
Total ways of choosing this =10C2*8C3*10C3=45*56*120=302400

c. 2 professor, 2 technical managers and 4 database expert
Total ways of choosing this =10C2*8C2*10C4=45*28*210=264600


Total ways = 403200+302400+264600 = 970200
Total ways is 970200.

Answer 2

we have 10 prof. (P),  8 tech. managers(TM),  and 10 DB experts (DBE).

we have to select at least  2 P, 2 TM, and 3 DBE.  That makes up a total of 7.  Then one more person can be a P, TM or DBE.

number of ways of formation of committee
     = ways of selecting 3 P +  2 TM + 3 DBE 
       + ways of selecting 2 P + 3 TM + 3 DBE
           + ways of selecting 2 P + 2 TM + 4 DBE

   = C(10,3) * C(8,2) * C(10,3)
        + C(10,2)* C(8,3) * C(10,3)
             + C(10,2) * C(8,2) * C(10,4)

   =   10*9*8/3! * 8*7/2!  * 10*9*8/ 3!
           + 10*9/2! * 8*7*6/3!  * 10*9*8/3!
                 + 10*9/2!  * 8*7/2!  *  10*9*8*7/4!

   = 560*720 + 420*720 + 45*28* 210
   =  970, 200