Solution

Can you believe it! The answer I got is 1152! Only eight children and they can arrange themselves in 1152 different ways! No, something must be wrong. Please let me know the correct answer.

But this how I arrived at 1152: The boys and girls can arrange themselves, with respect to the starting point "S", only in two ways: BBBBGGGG or GGGGBBBB, as I have already mentioned. This implies that although there are eight places available, the boys actually have four places to choose from to sit, and the girls also have only four places where they can sit.

But among themselves the boys can sit in any order and so can the girls. Let me take a simpler version of the question by considering only two boys - B1 and B2, and two girls - G1 and G2. Let's see how they can arrange themselves.

First let me consider the case where boys sit from the starting point. So the boys can sit in the order B1B2 or B2B1, and the girls can sit in the order G1G2 or G2G1. So all the possible arrangements are: B1B2G1G2, B1B2G2G1, B2B1G1G2, and B2B1G2G1; there are four possible arrangements. Similarly, if the girls chose to sit from the starting point, then there would be four arrangements: G1G2B1B2, G1G2B2B1, G2G1B1B2, and G2G1B2B1. Therefore, there are eight possible arrangements in all.

What if there are three boys - B1, B2, and B3, and three girls - G1, G2, and G3? Well, in that case the boys, among themselves, can arrange themselves in 3 x 2 x 1 = 6 ways and the girls, similarly, can sit in 3 x 2 x 1 = 6 ways. So the total number of arrangements will be 6 x 6 = 36 either way. Thus, if the boys were to sit from the starting point there will be 36 ways, and if the girls were to sit from the starting point there will be 36 ways. The total number of ways the children can arrange themselves, then, will be 36 + 36 = 72.

Now, I think I can work out the seating arrangements involving four boys and four girls. The four boys among themselves can sit in 4 x 3 x 2 x 1 = 24 ways. Similarly, the girls among themselves can sit in 4 x 3 x 2 x 1 = 24 ways. Therefore if the boys were to sit from the starting point then there would be 24 x 24 = 576 ways. On the other hand, if the girls were to sit from the starting point, there would again be 24 x 24 = 576 ways. Therefore, the total number of ways the children can sit = 576 + 576 = 1152 ways.

The picture shows boys sitting from the starting point. It shows all the 24 ways the boys can sit and all the 24 ways the girls can sit. Each arrangement for the boys (for example, B1B2B3B4) will correspond with 24 arrangements for the girls. Thus, the total number of arrangements will be 24 x 24 = 576. Similarly, if the girls were to sit from the starting point there would be 576 arrangements. The total number of ways is therefore 1152. Did I do that right?