prob question need help please (1 Viewer)

[cyl123]

Hey I was wondering if there is a short and easier way to do this question

How many ways can 6 boys and 4 girls be arranged in a row so that no 2 girls are together?

I know how to do it using a long and tedious method, which is finding total cases and subtracting number of cases 2 or more girls are together from it. If there is an easier and more effective way to approach this question, I would like to know.
Thanks

 

[NickP101]

Just out of interest is the answer 604 800? If it is i have an easy way, if its not well :S lol

 

[cyl123]

Yep it is but i know the easier method now
thanks anyway

 

[ianc]

would you care to enlighten us?

lol these questions kill me

 

[NickP101]

Line the 6 boys up first:

1 1 1 1 1 1

Then a boy must be between each girl (since no 2 girls can sit together). So the girls can sit in the spots numbered 2:

2 1 2 1 2 1 2 1 2 1 2 1 2

Hence boys can be arranged 6p6 and girls can be arranged 7p4.

 

[KFunk]

It's been a while since I've done the perms/combs thing but I think you should be able to do something along the lines of:

Line up the boys: B B B B B B

Then consider the spaces where girls can go: ₁ B ₂ B ₃ B ₄ B ₅ B ₆ B ₇

No. ways to arrange the boys = 6!

No. ways to pick spaces = ⁷C₄

No. ways to arrange girls in spaces = 4!

 

[ianc]

Thanks!

 

[cyl123]

Also just wondering, if the boys and girls were arranged in a circle instead, would you just change the 6! into 5! (ways of arranging boys in a circle) and change the 7P4 into 6P4 (as there is one less space)?

 

[Riviet]

Also just wondering, if the boys and girls were arranged in a circle instead, would you just change the 6! into 5! (ways of arranging boys in a circle) and change the 7P4 into 6P4 (as there is one less space)?

You would change it for the boys only as there will be a way to differentiate between positions on the circle once the boys are in their positions.

 

[cyl123]

But wouldnt there be one less space for the girls to be place if the boys were arranged in a circle, so you have to change for both boys and girls

 

[Riviet]

But wouldnt there be one less space for the girls to be place if the boys were arranged in a circle, so you have to change for both boys and girls

Yes, that's correct (after I drew a diagram ). Sorry my bad.

 

[Roobs]

out of interest, how do you do it the long and tedious way?