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

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Solution

$(2 x - 3 y + 4 z)^{9}$
Multinomial for $(x_{1} + x_{2} + . . . . . x_{k})^{n} = \sum (\begin{matrix} n n_{1} n_{2} - n_{k} \end{matrix}) {x_{1}}^{n_{1}} {x_{2}}^{n_{2}} {. . . . . . . . . x}_{k}^{k_{1}}$
Here, $x_{1} = 2 x - 3 y = x_{2}; 4 z = x_{3} k = 3, n = 9$
For the term containing $y, n_{1} = 0, n_{2} = 1, n_{3} = 0$
$∴$ Reg term $(\begin{matrix} 9010 \end{matrix}) (2 x)^{0} (- 3 y)^{1} (4 z)^{0}$
$= \frac{9!}{0! 1! 0!} \times (- 3 y)$
$= - 1088640 y$
$∴$ Coefficient of $= - 1088640$

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