Number of ways 6 boys and 6 girls. No 2 B or 2 G together

[jaus tail]

Homework Statement

Number of ways in which 6 boys and 6 girls can sit such that no 2 boys and 2 girls are together in a row.

Homework Equations

If there are n different objects in a row, then to place the we have n! ways
If n objects have n1 object of 1 kind then we have
n!/(n1)!

The Attempt at a Solution

we can have:
6 boys can sit in 6! ways
Now there are 7 spots and 6 girls so we have 7P6 (P because order matters as G1 then G2 is different from G2 then G1)
So we have 7P6 ways for girls to sit
So total number of ways is 6! times 7P6 ways which is 3628800
Book answer is 6! 6! + 6! 6! which is 1036800
Is book wrong?

Even from this link, I think book is right. But why can't we use 7P6?
GMAT Club:
https://gmatclub.com/forum/six-boys-and-six-girls-sit-in-a-row-at-random-find-the-57448.html

 

[Orodruin]

Is book wrong?

No, you are.

So we have 7P6 ways for girls to sit

No you do not. You need to think about if all of those (7) ways really results in no boys sitting together.

Edit: To make it clearer. Let us represents the boys with o and the "possible girl positions" with x. You have
xoxoxoxoxoxox
Can you take away any of the x and still have no two o together? (This is essentially your argument: "I can take away anyone x and nowhere will there be an xx or an oo." If this would be true you would indeed get 7 over 6 possibilities.)

 

[PeroK]

Homework Statement

Number of ways in which 6 boys and 6 girls can sit such that no 2 boys and 2 girls are together in a row.

Homework Equations

If there are n different objects in a row, then to place the we have n! ways
If n objects have n1 object of 1 kind then we have
n!/(n1)!

The Attempt at a Solution

we can have:
6 boys can sit in 6! ways
Now there are 7 spots and 6 girls so we have 7P6 (P because order matters as G1 then G2 is different from G2 then G1)
So we have 7P6 ways for girls to sit
So total number of ways is 6! times 7P6 ways which is 3628800
Book answer is 6! 6! + 6! 6! which is 1036800
Is book wrong?

Even from this link, I think book is right. But why can't we use 7P6?
GMAT Club:
https://gmatclub.com/forum/six-boys-and-six-girls-sit-in-a-row-at-random-find-the-57448.html

A suggestion: in these questions you can always check your answer for the same problem with a smaller number. In this case if you have two boys and two girls, then you can simply write down all the possibilities:

$B_1G_1B_2G_2$
$B_1G_2B_2G_1$

etc.

With your method, I think you would allow:

$G_1B_1 \ \ B_2G_2$

 

[jaus tail]

Oh yeah. I can't have
_G_G_G_G_G_G_G_
I can't have boys such that between any blank is empty.
Like no:
B G B G G B G B G B G B
So there aren't 7 places for girls to go. There are only 6 places with 2 possibilities
1) Leftmost seat is occupied by boy
2) Rightmost seat is by boy.