Even if we aren't given the actual names in the problems, everyone does have a name. So when the problem says "the boys finish in alphabetical order based on their names" this means (as far as solving the problem is concerned) that the order of the boys is fixed and already known.

It looks like your solution wants to arrange the boys first, and then arrange the girls. Since the order of the boys is fixed, the $6!$ is not needed. You are then correct that there are $7$ spaces for the first girl. However, the problem doesn't say that girls can't finish next to each other, so there are actually $8$ spaces for the second girl. This pattern continues giving an answer of $1\cdot 7\cdot 8\cdot 9\cdot 10$ which matches the answer given.