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One of the possible solution is taking $X_1,X_2,X_3,\ldots,X_r$ as boxes and no. of objects in these boxes as $0\leq X_1\leq X_2\leq X_3\leq \cdots\leq X_r$ where $[X_1+X_2+X_3+\cdots+X_r=N]$. But how to find this? Some more Methods may also be there.

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Welcome to math.SE, bittuhere! Could you clarify your question a bit? Are you asking for the number of distinct unordered ways to distribute the objects in the boxes? If so, this is the same as the number of 'partitions' of the number $N$ into $R$ summands. The question… might be useful. – Paul VanKoughnett Mar 14 '13 at 6:47
You seem to be looking for partitions of the number $N$ into $R$ or lesser parts. The reference above should then help. – Macavity Mar 14 '13 at 6:54

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