Distribution with restrictions [S]

When I wrote this post I made a typo mistake. Here is what I intended.
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Problem: Enumerate the number of solutions of the equation



such that for every

Lets suppose so that at least one solution is assured.
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The flipside of this is simple. How many solutions with each
? Easy.. put balls in each of the boxes, a priori. Now distribute rest in :)
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solution: Suppose

Let

(i) only one of the entries has size

and rest are less than that. Now we get the regular inclusion-exclusion expression

(ii) For two,

and so on.

Summing up individually we get

Hence the number of solutions with the given restriction is