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Sum of multinomial coefficients

MediumCombinatorics~4m

Compute the sum of all multinomial coefficients (7b1,b2,b3,b4)\dbinom{7}{b_1, b_2, b_3, b_4} over non-negative integers with b1+b2+b3+b4=7b_1 + b_2 + b_3 + b_4 = 7.

Show hints (2)+
  1. Multinomial theorem: set all variables to 11 in (x1+x2+x3+x4)7(x_1+x_2+x_3+x_4)^7.
  2. That gives 474^7.

Answer

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16384

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Asked at: Jane Street, Citadel

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