The 12 choose 7 is exactly like the 14 choose 5. You seem to be using the "stars and bars" method. In other words, the different distributions of skateboards look like **|***|*|*|| where this would mean child 1 gets 2 skateboards, child 2 gets 3, child 3 gets 1, child 4 get 1, and child 5 and 6 get 0 skateboards.

Hence to count the number of ways to distribute, you have 12 symbols (7 *'s and 5 bars) and you have to choose what positions to place the 5 bars.. hence 12 choose 5. Alternative you can do 12 choose 7 = 12 choose 5 which would correspond to choosing what positions to put the 7 *'s instead.

If they multiplied correctly then the answer is correct.