Out of 5 women and 4 men, a committee of three members is to be formed in such a way that at least one member is a women. In how many different ways can it be done ?

Solution (By Examveda Team)

Required number of ways
$$\left( {{}^5{{\text{C}}_1} \times {}^4{{\text{C}}_2}} \right) + \left( {{}^5{{\text{C}}_2} \times {}^4{{\text{C}}_1}} \right)$$     $$ + \left( {{}^5{{\text{C}}_3}} \right)$$
$$ = \left( {5 \times \frac{{4 \times 3}}{{2 \times 1}}} \right)$$   $$ + \left( {\frac{{5 \times 4}}{{2 \times 1}} \times 4} \right)$$   $$ + \left( {\frac{{5 \times 4 \times 3}}{{3 \times 2 \times 1}}} \right)$$
$$\eqalign{ & = \left( {30 + 40 + 10} \right) \cr & = 80 \cr} $$