Front-loaded Boxes

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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