If x boys and y girls sit in a row randomly, in how many ways can all the girls sit together ?

144
200
240
260

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.