Problem Wih Probability....

[Cat-astrophe]

Hey guys!
I have a doubt in the logic of a rather fundamental problem in probability---

You have 5 boys and 5 girls , how many ways can you arrange them such that no boy sits next to a girl?

The std appraoch is that you place the 5 boys in 5! ways and then the girls in 6P5-- 6! ways as there are 6 vacancies.. so ur answer is 5!6!

Whats bothereing moi si that - when u seat 5 girls with a choice of 6 seats, you have one that is vacant right??

So one of the arrangements that you are counting is- where the space between two boys is vacant- and thus you do have two guys sitting next to each other?(gb bg bg bg bg )
shudn't you reduce that from 5!*6!??

Is there a major fundamental kink in my reasoning? Waht am i missing?

Enlighten me!

[mizzou]

u sure that question is :
"You have 5 boys and 5 girls , how many ways can you arrange them such that no boy sits next to a girl? " ?

if it is so then 5!6! is wrong ans.

5!6! is correct when question is "You have 5 boys and 5 girls , how many ways can you arrange them such that no girls sit together? "

For the former question, i think ans is 0.

[Cat-astrophe]

YUPPPP! Thats right .. i meant ur question- sorry!
but am not convinced abt, 5!6!

[zango]

Originally Posted by Cat-astrophe

YUPPPP! Thats right .. i meant ur question- sorry!
but am not convinced abt, 5!6!

Seat the boys in 5! ways - then you have 6 places where you can seat 5 girls, such that no girls are together : xbxbxbxbxbx (x denotes places where the girls MAY be seated). So, you can choose 5 of these 6 in 6C5 ways, and then, have 5! arrangements - incidentally, this is the same as 6P5 = 6!
Therefore, you have 5!6! total number of ways.

HTH.

[newguy]

Originally Posted by Cat-astrophe

YUPPPP! Thats right .. i meant ur question- sorry!
but am not convinced abt, 5!6!

What is your question ?

you don't want any girl to sit together OR you don't want any two boys as well as two girls to sit together.

Well, in case your question is the former one, be content with 5!6!

ELSE

there will be only 5 places where girls can go once boys have settled. So, the answer would be 5!5!.

[zango]

Originally Posted by newguy

What is your question ?

you don't want any girl to sit together OR you don't want any two boys as well as two girls to sit together.

Well, in case your question is the former one, be content with 5!6!

ELSE

there will be only 5 places where girls can go once boys have settled. So, the answer would be 5!5!.

No, that is incorrect. if the question is the second one, then it will be 2*5!5! and not 5!5!
This is because there are two possible arrangementS:
bgbgbgbgbg or gbgbgbgbgb
and hence the answer will be 2 * 5! * 5!

[newguy]

Originally Posted by zango

No, that is incorrect. if the question is the second one, then it will be 2*5!5! and not 5!5!
This is because there are two possible arrangementS:
bgbgbgbgbg or gbgbgbgbgb
and hence the answer will be 2 * 5! * 5!

OOPS ! Thanks Buddy...I did the Ladies First thingie !

[Cat-astrophe]

Hi thanks all!

But am a lil confused still.. wud appreciate if someone cud demistify me-

Abt. the question that would give me the ans as 5!6!, what i dont understand is that if you are to seat 5 girls in 6 ways, after arranging the girls you'd still have one space vacan right??

what if that vacant space is between two boys?? wouldn't that be 2 boys sitting together?

[newguy]

Originally Posted by Cat-astrophe

Hi thanks all!

But am a lil confused still.. wud appreciate if someone cud demistify me-

Abt. the question that would give me the ans as 5!6!, what i dont understand is that if you are to seat 5 girls in 6 ways, after arranging the girls you'd still have one space vacan right??

what if that vacant space is between two boys?? wouldn't that be 2 boys sitting together?

The entire thread had been discussing this only..read this thread ( my entries )..you will understand !

[jazztheroof]

tk this approach
1st u put boys in e row with one blank space in between each of them ( this is to make sure gals hv to sit individually only )
as there r 5 boys so it will look lik this

_b_b_b_b_b_
_ denotes places where the girls MAY be seated

this means wen 5 boys r sitting lik this they r givin 6 individual vacant spaces for the gals to sit ............. in this case no 2 boys r sitting together .
when we put 5 gals in any 5 out of 6 vacant spaces no 2 gals will b sitting together
so gals can choose 5 of these 6 vacant spaces in 6C5 = 6!

and as far as ur doubt comprehend lik this
in any case gals occupie 5 places , lving one space vacant but still no 2 boys will sit together as there is e vacant space in between them but mind u this ques is not talking abt whether boys can sit together or not , it only speaks abt gals

and the ans 5!6! is strictly for the ques, which is
You have 5 boys and 5 girls , how many ways can you arrange them such that no girls sit together?

gOdSpeEd