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## #1 2012-12-17 03:35:52

jacks
Member
Registered: 2012-11-21
Posts: 128

### boys and girls

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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## #2 2012-12-17 03:49:35

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,410

### Re: boys and girls

Hi;

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

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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## #3 2012-12-17 03:56:01

jacks
Member
Registered: 2012-11-21
Posts: 128

### Re: boys and girls

Yes bobbym you are right.

But where i have done mistake in my solution.

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## #4 2012-12-17 04:06:22

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,410

### Re: boys and girls

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!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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## #5 2012-12-17 04:11:10

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: boys and girls

jacks wrote:

But where i have done mistake in my solution.

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.

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## #6 2012-12-17 05:42:01

jacks
Member
Registered: 2012-11-21
Posts: 128

### Re: boys and girls

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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## #7 2012-12-17 12:07:45

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,410

### Re: boys and girls

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

Not at the same time!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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## #8 2012-12-17 12:10:53

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: boys and girls

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.

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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## #9 2012-12-17 12:14:42

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,410

### Re: boys and girls

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.
No great discovery was ever made without a bold guess.

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## #10 2012-12-17 12:15:45

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: boys and girls

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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## #11 2012-12-17 12:19:42

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,410

### Re: boys and girls

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.
No great discovery was ever made without a bold guess.

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## #12 2012-12-17 12:30:56

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: boys and girls

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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## #13 2012-12-17 12:37:13

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 106,410

### Re: boys and girls

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.
No great discovery was ever made without a bold guess.

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