Duck Distribution | Math Problem

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Could anyone solve this? don’t give me the answer, I just wanna understand the problem

Well this is basically a pigeonhole principle problem.

You have to maximise number of ducks. => minimise number of repeated cards to make room for more ducks

Now think, when picking 7 boxes, how many boxes do you exclude?
12 - 7 = 5 boxes right? Therefore by PGP, one ducks card must be in at least 5+1=6 boxes to be available to 7 picked boxes. (crux move)

I think that’s enough hint, I’ll let you solve the rest.

ANSWER BELOW :

6968