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The answer is given by ((nr))=(n+r−1r−1) sometimes called n "multichoose" r
Take n=100−3 and r=3 and then simplify normally. (Effectively only 97 objects need to be distributed)
Even the proof is pretty simple. Consider dividing n identical objects into r groups using r−1 identical partitions.
The number of arrangements gives the number of solutions. Which simplifies to n multichoose r
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