There are 10 girls and 8 boys in a class room including Mr. Ravi, Ms. Rani and Ms. Radha. A list of speakers consisting of 8 girls and 6 boys has to be prepared. Mr. Ravi refuses to speak if Ms. Rani is a speaker. Ms. Rani refuses to speak if Ms. Radha is a speaker. The number of ways the list can be prepared is a 3 digit number `n_(1)n_(2)n_(3)`, then `|n_(3)+n_(2)-n_(1)|=`

Options

A. 202
B. 308
C. 567
D. 952

Explanation

Following cases arise:
CASE I When Ms. Radha is a speaker
In this case, Ms. Rani refuses to speak. So, we have to select 7 girls out of remaning 8 girls and 6 boys from 8 boys. The numbr of ways of doing this is `""^(8)C_(7)xx""^(8)C_(6)=224`.
CASE II When Radha is not a speaker
In this case, two subcases arise.
(i) When Rani is a speaker (ii) When Rani is not a speaker.
If Rani si a speaker, Mr Ravi refused to speak. So, we have to select 7 girls from remaining 8 girls and 6 boys from remaining 7 boys which can be done in `""^(8)C_(7)xx""^(7)C_(6)` ways = 56 ways.
If Rani is not a speaker, then 8 girls are to be chosen out of remainig 8 girls and 6 boys from 8 boys which can be done in `""^(8)C_(6)xx""^(8)C_(8)=28` ways.
Hence, required number of ways =224+56+28=308.