Question

From six gentlemen and four ladies, a committee of five is to be formed. Number of ways in which this can be done if the committee is to include at least one lady is:

• A. $252$
• B. $246$
• C. $248$
• D. $250$

Answer 1

Answer: (B)

$246$

We can select $5$ members for the committee to include at least one lady in the following four ways as done below:
(1) $1$ lady and $4$ gentlemen
(2) $2$ ladies and $3$ gentlemen
(3) $3$ ladies and $2$ gentlemen
(4) $4$ ladies and $1$ gentlemen
Hence, the required number of committees
$={}^4{C}_1\times {}^6{C}_4+{}^4{C}_2\times {}^6{C}_3+{}^4{C}_3\times {}^6{C}_2+{}^4{C}_4\times {}^6{C}_1$
$=60+120+60+6=246$