Number of Ways to Seat Boys and Girls

To find the number of ways in which 6 boys and 6 girls can be seated in a row such that no two girls sit together, we can follow the steps below:

Step 1: Arrange the boys
Since there are 6 boys, we have 6 factorial (6!) ways to arrange them in a row. This gives us a total of 720 possible arrangements of boys.

Step 2: Create spaces for girls
To ensure that no two girls sit together, we need to create spaces between the boys where the girls can be seated. We can do this by considering the following cases:

Case 1: Girls sit at the ends
In this case, we have a boy at each end and 4 boys in the middle. There are 5 possible spaces where the girls can sit (2 ends and 3 spaces between the boys).

- First, we select 6 spaces out of the 5 available spaces for the girls to sit. This can be done in 5 choose 6 ways (5C6).
- Next, we arrange the 6 girls in the selected spaces. This can be done in 6! ways.
- Therefore, the total number of arrangements in this case is 5C6 * 6!.

Case 2: Girls sit in the middle
In this case, we have 6 boys at the ends and 4 spaces in between them where the girls can sit. There are 5 possible spaces where the girls can sit (4 spaces between the boys and 1 space after the last boy).

- First, we select 6 spaces out of the 5 available spaces for the girls to sit. This can be done in 5 choose 6 ways (5C6).
- Next, we arrange the 6 girls in the selected spaces. This can be done in 6! ways.
- Therefore, the total number of arrangements in this case is 5C6 * 6!.

Step 3: Calculate the total number of arrangements
To find the total number of arrangements, we add the number of arrangements from Case 1 and Case 2.

Total number of arrangements = (5C6 * 6!) + (5C6 * 6!)

Simplifying the expression, we get:

Total number of arrangements = 2 * (5C6 * 6!)

Final Answer:
The number of ways in which 6 boys and 6 girls can be seated in a row such that no two girls sit together is 2 * (5C6 * 6!) or 2 * 5! * 6!.

36 ways