Distributing candies with a cap

MediumCombinatorics~5m

In how many ways can 1010 identical candies be distributed among 44 children if no child may receive more than 44 (a child may still get none)?

Show hints (2)+
  1. Start with unrestricted stars and bars, then remove the over-cap cases.
  2. Inclusion–exclusion: subtract "one child 5\ge 5", add back "two children 5\ge 5".

Answer

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68

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Asked at: Jane Street, Optiver

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