Six boxes labeled $$1,2, \ldots, 6$$ are arranged in a line. Seven identical balls are to be placed into the boxes such that for any $$1 \leq k \leq 6,$$ there are at least $$k$$ total balls amongst boxes $$1,2 , \ldots k.$$ How many different placements of the balls into the boxes are possible?

Details and assumptions

All 7 balls have to be placed.

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