Answer
2522520
Work Step by Step
Let's number the children from oldest to youngest to make this problem easier. Then, we realize that this is just a sorting of the cones, $14!$. However, each type of cone is indistinguishable from another of its type. Therefore, we divide by repetitions, namely $3!$ for vanilla, $2!$ for chocolate, $4!$ for strawberry, and $5!$ for butterscotch.
$\frac{14!}{3!*2!*4!*5!} = 2522520$