A committee of 5 members is to be formed out of 3 trainees, 4 professors and 6 research associates. In how many different ways can this be done, if the committee should have 4 professors and 1 research associate or all 3 trainees and 2 professors?

Solution (By Examveda Team)

Required number of ways
$$ = \left( {{}^4{C_4} \times {}^6{C_1}} \right) + $$   $$\left( {{}^3{C_3} \times {}^4{C_{2}}} \right)$$
$$\eqalign{ & = \left( {1 \times 6} \right) + \left( {1 \times \frac{{4 \times 3}}{2}} \right) \cr & = \left( {6 + 6} \right) \cr & = 12 \cr} $$