Answers
First, there are 11 different possible boxes, 200, 201, etc, all the way up to 210
To guarantee at least two boxes with the same number of apples, you need 12 boxes, because otherwise it is possible, no matter how unlikely, that you merely have one of each, no duplicates.
To see how many are necessary to guarantee a triplication, consider that there are also 11 possible pairs, that is, configurations where two are the same. Since it is possible, with 22 boxes, that you have 11 pairs, not one triplicate, it takes 23 boxes to guarantee one.
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