Each group will receive one pie and one apple per child.
Each group will receive one pie, and one apple per child.
In the first case each group receives many apples and many pies. In the second case each group receives many apples, but each group receives only one pie.
I need to describe the rule that makes it so the comma changes the meaning in this sentence, but I'm not sure what the rule operating here is called, nor what the relationship between these two sentences is called.
(While this is about the use of the comma, I strongly suspect the comma here is delineating two different syntactic reasonings resulting in the same words in the same order, so the grammar tag has been included. If you object to this use of the grammar tag, you can remove it. Just at least think about it first :P )