In how many ways can 4 girls and 3 boys sit in a row such that just the girls are to sit next to each other? Answer: 288

Question

In how many ways can 4 girls and 3 boys sit in a row such that just the girls are to sit next to each other? Answer: 288

Please explain how to get this.

I understand that we have

GGGG => 4 girls next to each other B B B => 3 boys

but how do you put them together and work out the number of possible ways. They are different so not identical

Answer 1

We can have:

GGGGBBB, BGGGGBB, BBGGGGB, BBBGGGG

(There are four ways to place a group of four consecutive girls in a row of seven.

The girls can be permuted in each case $4!$, and so can the boys $(3!)$.

$$\bf 4\times 4!\times 3! = 576$$

NOTE If the intention of the author was that girls must sit next to a girl, and boys next to a boy, then there are only two ways to place the group of girls: GGGGBBB, BBBGGGG.

In that case, we have $2 \times 4!\times 3! = 288$.

Answer 2

To be honest I am not sure if this is the right way to do it, some one please correct me if Im wrong.

By thinking about the group of girls and boys separately, (as mentioned in the comment):

girls = $\ G_1,G_2,G_3,G_4$; boys = $\ B_1,B_2,B_3$

we can create $\ 4$! ways of arranging the girls(by themselves) and 3! possibilities of arranging the boys by themselves.

now if we put them together, we are doubling the possibilites of arrangement.

thus, # of arrangements = $\ 4!*3! *2 =288 $

Answer 3

I doubt whether you have worded your question correctly as according to your question the answer should come out as 576 and not 288. Anyway an approach that could lead to 288 is as follows:

Let's number girls and boys as : G1, G2, G3, G4 and B1, B2, B3 respectively.

Since a girl has to sit next to a girl only therefore all the girls would sit together G1,G2,G3,G4 or G2,G3,G1,G4 etc. Now these girls can be arranged in 4! = 24 ways.

Now we have 3 boys who can be made to sit together in 3! = 6 ways.

Finally we have Girls (G) and Boys (B) who have to be seated together and they can be seated in 2! = 2 ways (consider 4 girls as one group = G and 3 boys as one group = B)

From above three steps we get the total number of ways as : 4! * 3! * 2! = 288

Answer 4

I can explain why this answer is 288 and not 576. The difference between it being 288 and 576 is based on one word in the question: "just"

If the boys and girls are arranged where just the girls (and not the boys) are sitting next to each other, then the problem would be: 2 x 4! x 3! which = 288 instead of 4 x 4! x 3! which is 576.

Why 2 instead of 4? If you have three boys 1 2 3, then there's 2 spaces the girls could fit within them (between the 1 and 2, and between the 2 and 3). If the girls are placed before the 1 or after the 3, then it allows the boys to automatically be together which the question says just the girls should sit next to each other.

I hope that explains it. My text book had 2 variations of the same question where one was "In how many ways can they sit in a row if the girls are to sit together?" and "In how many ways can they sit in a row if just the girls are to sit together. Both questions had different answers like the ones above.