Solution

Method 1 – Inclusion-Exclusion Principle

Intuition

The problem asks for the number of ways to distribute n candies among 3 children, with each child getting at most limit candies. If there were no upper limit, the answer would be the number of non-negative integer solutions to a + b + c = n, which is C(n+3-1, 3-1) = C(n+2, 2). The limit constraint can be handled using the inclusion-exclusion principle: subtract cases where one or more children exceed the limit.

Approach

Count all non-negative solutions to a + b + c = n (no limits): use the stars and

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