There are    ways to select two children of 10 for vanilla ice cream;

There are    ways to select three children of remaining 8 for chocolate cones;

There are    ways to select four children of remaining 5 for strawberry cones;

and there is just NO CHOICE (the unique selection) for this lucky one who will get the pistachio cone.


Now take the product  ..  to get your answer.

It is the number of all distinguishable permutations of 10 objects, two of which are indistinguishable vanilla ice cream;


the other three are indistinguishable chocolate cones, and the other four are indistinguishable strawberry cones.


In all, there are   such distinguishable arrangements.