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Find the coefficient of $x^3y^2z^3w$ in the expansion of $(2x+3y-4z+w)^9$

Using the formula of multinational coefficients $$ \begin{pmatrix} n \\ r_1,r_2,...,r_k \\ \end{pmatrix}= \ \frac{n!}{r_1! \cdot r_2! \cdot \ldots \cdot r_k!} \ $$

$$\Rightarrow \begin{pmatrix} 9 \\ 3, 2, 3, 1 \\ \end{pmatrix}= \frac {9!}{3!2!3!1! } = \bbox[yellow]{5040} $$

Would the above be the answer or does one have to take this one step further by further expanding?

$$ 5040 *(2)(3)(-4)(1) = -120960$$

Rodrigo de Azevedo
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Jon
  • 1,920

1 Answers1

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$5040 \times 2^3 \times 3^2 \times (-4)^3 \times 1 =-23224320$

ThePortakal
  • 5,158