Question

The number of ways in which a committee of $3$ ladies and $4$ gentlemen can be appointed from a meeting consisting of $8$ ladies and $7$ gentlemen, if Mrs $X$ refuses to serve in a committee if Mr. $Y$ is a member is

• A. 1960
• B. 1540
• C. 3240
• D. none of these

Answer 1

Answer: (B)

1540

$3$ ladies out of $8$ can be selected in ${}^8{C}_3$ ways and $4$ gentlemen out of $7$ in ${}^7{C}_4$ ways.

Now each way of selecting $3$ ladies is associated with each way of selecting $4$ gentlemen.

Hence, the required number of ways ${=}^8{C}_3{\times }^7{C}_4=56\times 35=1960$.

We now find the number of committee of $3$ ladies and $4$ gentlemen in which both Mrs $X$ and Mr $Y$ are member.

In this case we can select $2$ other ladies from the remaining $7$ in ${}^7{C}_2$ ways and $3$ other gentlemen from the remaining $6$ in ${}^6{C}_3$ ways.

$\therefore$ The number of ways in which both Mrs $X$ and Mr $Y$ are always included ${=}^7{C}_2{\times }^8{C}_3=21\times 20=420$

Hence the required number of committee in which Mrs $X$ and Mr $Y$ do not serve together $=1960=420=1540$.