Question

From $6$ gentlemen and $4$ ladies a committee of $5$ is to be formed. Number of ways can this be done if
(a) The committee is to include at least one lady.
(b) there is no restriction about its formation.

• A. $240,255$
• B. $242,256$
• C. $246,252$
• D. none of these

Answer 1

Answer: (C)

$246,252$

(a) There should atleast be one lady, i.e., (L,G)$\equiv (1,4),(2,3),(3,2),(4,1)$
So, we will choose L ladies from 4 and G gentlemen from 6,
${=}_1^4\text{C}{.}_4^6\text{C}{+}_2^4\text{C}{.}_3^6\text{C}{+}_3^4\text{C}{.}_2^6\text{C}{+}_4^4\text{C}{.}_1^6\text{C}=60+120+60+6=246$
(b)There is no restrictions
${=}_5^{10}\text{C}=252$
Hence, option 'C' is correct.