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**jacks****Member**- Registered: 2012-11-21
- Posts: 80

6 boys and 6 girls are sitting in a row . Then the no. of ways that boys and girls sits alternatively

my solution::

Using gap method::

here X = denote boys and _ = denote girl

Then X _ X _ X _ X _ X _ X _

first we can arrange boys , which can be done in 6! ways

now arrange girls in 7 gaps ,

for that first we select 6 place out of 7 which can be done in C (7,6) ways and now arrange these 6 girls

Which can we done in C(7,6) * 6! = 7!

So total no. of ways is = 6! * 7!

but answer is = 2*6! *6!

so where i have done mistake

Thanks

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,589

Hi;

I am getting 2 * 6! * 6! as the answer. Provided we are assuming the girls and boys are distinct which is reasonable.

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**jacks****Member**- Registered: 2012-11-21
- Posts: 80

Yes bobbym you are right.

But where i have done mistake in my solution.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,589

Arrange the girls first in a line;

_ G _ G _ G _ G _ G _ G _ = 6!

In the first _ 6 boys can go

_ G (6 boys ) G _ G _ G _ G _ G _

In the second 5 boys

_ G (6 boys ) G (5 boys ) G _ G _ G _ G _

all the way down to

_ G (6 boys ) G (5 boys ) G (4 boys) G (3 boys) G (2 boys) G _

That can be done in 6 x 5 x 4 x 3 x 2 = 6!

So far we have 6! * 6!, now the last two slots

_ G (6 boys ) G (5 boys ) G (4 boys) G (3 boys) G (2 boys) G _

the one remaining boy can go in 2 ways.

2 * 6! * 6!

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**If it ain't broke, fix it until it is.**

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

jacks wrote:

But where i have done mistake in my solution.

Your mistake is in

jacks wrote:

now arrange girls in 7 gaps ,

You can't do that as that allows for possibilities in which there's a girl at both ends, which you don't want. The possibilities are either

X_X_X_X_X_X_

or

_X_X_X_X_X_X

Each set has

ways; hence there are ways altogether.Offline

**jacks****Member**- Registered: 2012-11-21
- Posts: 80

Thanks bobbym and scientia.

but i did not understand the meaning of

You can't do that as that allows for possibilities in which there's a girl at both ends

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,589

You can't do that as that allows for possibilities in which there's a girl at both ends

Not at the same time!

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

Well, maybe the girls at the ends are in a superposition and our observation force the situation to enter one of the two possible states.

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Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,589

Point is, sometimes it is a guy and sometimes it is not. Depending on what slot the last guy goes into.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

I was just kidding!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,589

Hmmmm. I know that. I know that you kid but you do not joke.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

bobbym wrote:

You can't do that as that allows for possibilities in which there's a girl at both ends

Not at the same time!

Exactly. Jacks's method allows girls to be at boths ends at the same tiem that was where his proof went wrong.

He wanted us to tell him where his proof went wrong, didn't he?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,589

Hi;

I agree.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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