Balls into bowls, three each

In how many ways can 77 identical white balls and 88 identical black balls be distributed into 55 distinct bowls so that each bowl receives exactly 33 balls?

Show hints (2)+
  1. Each bowl's makeup is fixed by its white count wi{0,1,2,3}w_i\in\{0,1,2,3\}; need wi=7\sum w_i = 7.
  2. Capped stars-and-bars: (114)5(74)=330175\binom{11}{4} - 5\binom{7}{4} = 330 - 175.

Answer

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155

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Asked at: Jane Street, Citadel

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