Basically if the things your dividing it into are the same, i.e. if you swap them around it won't change your number of combinations. Also each pile/child must receive an equal amount to divide by m!.

For example the pile one you divide by 4! as there are 4 piles which are the same so after you distribute the presents you divide by 4! as the piles are the same. However with the children one, since you can also arrange the children then you wouldn't divide.

E.g. you have 2 piles, one with an apple and one with an orange, if you swap the apple and orange you still have one pile with an apple and one with an orange so it's basically a repeat, AO and OA so dividing by 2! removes the repetitions. If Billy has the apple and John has the orange, and if they swap them, then you have a new combination with Billy with the orange and John with the apple.

Hope this makes sense

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