Question

The number of ways in which a committee of 3 ladies and 4 gentlemen can be appointed out of 8 ladies and 7 gentlemen. If Mrs. X refuses to serve in a committee of which Mr Y is a member, is
$
  {\text{A}}{\text{. 1960}} \\
  {\text{B}}{\text{. 1540}} \\
  {\text{C}}{\text{. 3240}} \\
  {\text{D}}{\text{. None of these}} \\
$

Answer 1

Hint: In this question, we need to use combinations to get the required expressions. We will firstly find the number of ways in which the committee can be formed when Mr. Y is included and then the number of ways in which the committee can be formed when Mr. Y is not included. Then taking the sum of both cases will help us reach the answer.

Complete step-by-step solution -

We have been given that Mrs. X refuses to serve in a committee of which Mr Y is a member.
So, When Mr. Y is included then Mrs. X is not there.
So, the number of ways$ = {}^7{C_3} \times {}^6{C_3} = 35 \times 20 = 700$
Now, if Mr. Y is not a member then Mrs. X may be a member.
So, the number of ways$ = {}^8{C_4} \times {}^6{C_4} = 56 \times 15 = 840$
Therefore, the total number of ways in which committee can be formed is 700 + 840 = 1540 ways.

Note: Whenever we face such types of problems the value point to remember is that we need to have a good grasp over combinations and its formulas. The most basic formula to calculate combinations has been discussed above and used to solve the given question.