There are $8$ boys and $6$ girls. In how many ways can they line up so that the front and end of the line are occupied by boys?
Someone told me that the answer is $3832012800$. Am I missing something?
Would the solution not be:
We first choose the front and end of the line. Since they must both be occupied by boys, we have $8 \cdot 7$ possibilities. Then, there are $12$ people left, with no constraints for any of them. Thus, $8*7*12!$.
EDIT: I explained my solution, which I have been told is correct.