Tgere are 10 girls and 8 boys in a class.a list of speakers consisting of 8 girls and 6 boys is to be prepared.mr. ravi refuses to speak if Rani speaks.And Raani refuses to speak if Radha speaks.Find no. Of ways list can be made ?

If we were to have no personality conflicts among the speakers, we'd have 2 combinations multiplying against each other (one for the girls and the other for the boys):

#C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!)# with #n="population", k="picks"#

#((10),(8))((8),(6))=(10!)/((8!)(2!))(8!)/((6!)(2!))=(10!)/((6!)(2!)^2)=1260#

This is our upper limit - any answer bigger than this tells us we've made a mistake.

To figure out our question, we need to handle each case separately. I'll find the acceptable number of ways the list can be made under each set of assumptions and sum it all up at the end.

Our limitations arise if Rani or Radha speak and so we'll have 3 cases - neither speaks, Rani speaks, Radha speaks:

Neither Rani nor Radha

We're excluding Rani and Radha from the calculations, which means the girls have a population of 2 less - or in other words, but for Rani and Radha they all speak:

#((8),(8))((8),(6))=(8!)/((8!)(0!))(8!)/((6!)(2!))=(8!)/((6!)(2!))=28#

Radha speaks

If Radha speaks, Rani won't. That reduces the number of girls by 1:

#((9),(8))((8),(6))=(9!)/((8!)(1!))(8!)/((6!)(2!))=(9!)/((6!)(2!))=126#

Rani speaks

If Rani speaks, Ravi won't. That reduces the number of boys by 1:

#((10),(8))((7),(6))=(10!)/((8!)(2!))(7!)/((6!)(1!))=(10!)/(8xx(6!)(2!))=315#

In total

#28+126+315=469#