The correct answer is (c) 240.
There are $x+y$ people in total, and we want to count the number of ways to arrange them in a row such that all the girls sit together. We can do this by first considering the girls as a single unit. There are $y!$ ways to arrange the girls, and then there are $x!$ ways to arrange the boys. So the total number of ways is $y!x!$.
However, this overcounts the number of arrangements, since we don’t care about the order of the girls within their group. To account for this, we divide by $y!$. This gives us the number of ways to arrange the boys and girls such that all the girls sit together:
$$\frac{y!x!}{y!} = x!$$
For example, if there are 3 boys and 2 girls, then there are $2!3!$ ways to arrange them in a row:
- GGBB
- GBGB
- BGB
- BGG
However, only the first arrangement satisfies the condition that all the girls sit together. So the answer is $3!$, or 6.
The other options are incorrect because they do not account for the fact that we don’t care about the order of the girls within their group.