Examveda

From a group of 7 men 6 women, 5 persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

A. 564

B. 645

C. 735

D. 756

E. None of these

Solution(By Examveda Team)

Required number of ways
$$= \left( {{}^7{C_3} \times {}^6{C_2}} \right) +$$   $$\left( {{}^7{C_4} \times {}^6{C_1}} \right) +$$   $$\left( {{}^7{C_5} \times {}^6{C_0}} \right)$$
$$= \left\{ {\frac{{7 \times 6 \times 5}}{{3!}} \times \frac{{6 \times 5}}{{2!}}} \right\}$$     $$+ \left( {{}^7{C_3} \times {}^6{C_1}} \right)$$   $$+ \left( {{}^7{C_2} \times 1} \right)$$
$$= \left\{ {\frac{{7 \times 6 \times 5}}{6} \times \frac{{6 \times 5}}{{2 \times 1}}} \right\}$$     $$+ \left( {\frac{{7 \times 6 \times 5}}{{3 \times 2 \times 1}} \times 6} \right)$$    $$+ \left( {\frac{{7 \times 6}}{{2 \times 1}} \times 1} \right)$$
\eqalign{ & = \left( {525 + 210 + 21} \right) \cr & = 756 \cr}