Question

From 4 gentlemen and 6 ladies a committee of five is to be selected. The number of ways in which the committee can be formed so that gentlemen are in majority is

• A. 66
• B. 156
• C. 60
• D. none of these

Answer 1

The correct option is A 66

Let's represent gentlemen with $G$ and ladies with $W$.
Hence there are $4G$ and $6W$.
$G$ will be in majority in following combinations: $4G,1W;3G,2W$
Hence number of ways this can happen =$\left(\begin{array}{c}4\\ 4\end{array}\right)\left(\begin{array}{c}6\\ 1\end{array}\right)+\left(\begin{array}{c}4\\ 3\end{array}\right)\left(\begin{array}{c}6\\ 2\end{array}\right)=66$