Balls into bowls, three each

HardCombinatorics~8m

In how many ways can $7$ identical white balls and $8$ identical black balls be distributed into $5$ distinct bowls so that each bowl receives exactly $3$ balls?

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Each bowl's makeup is fixed by its white count $w_i\in\{0,1,2,3\}$ ; need $\sum w_i = 7$ .
Capped stars-and-bars: $\binom{11}{4} - 5\binom{7}{4} = 330 - 175$ .

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Balls into bowls, three each

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