Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**jacks****Member**- Registered: 2012-11-21
- Posts: 77

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

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,362

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

Offline

**jacks****Member**- Registered: 2012-11-21
- Posts: 77

Yes bobbym you are right.

But where i have done mistake in my solution.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,362

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

Offline

**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: 77

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

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,362

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

Offline

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,507

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

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,362

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

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

Offline

**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,507

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

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,362

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

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

Offline

**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?

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,362

Hi;

I agree.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

Offline

Pages: **1**