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Coefficient of (x^{3}y^{4}z^{2}) in (2x-3y+4x)^{9}

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Coefficient of $(x^{3} y^{4} z^{2})$ in $(2 x - 3 y + 4 x)^{9}$

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Coefficient of

(x^{3} y^{4} z^{2})

(2 x - 3 y + 4 x)^{9}

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Text solutionVerified

(2 x - 3 y + 4 z)^{9} = [(2 x - 3 y) + 4 z]^{9}

General term,

T_{rH} =^{9} C_{r} (2 x - 3 y)^{r} (4 z)^{9 - r}

General term for

(2 x - 3 y)^{r} =^{r} C_{s} (2 x)^{s} (- 3 y)^{r - s}

\Rightarrow T_{rH} =^{s} C_{r} s = 0 \sum r^{r} C_{s} (2 x)^{s} (- 3 y)^{r - s} (4 z)^{9 - r}

For

x^{3} y^{4} z^{2}, 9 - r = 2 \Rightarrow r = 7, s = 3

\Rightarrow

Coefficient

=^{9} C_{7} \cdot^{7} C_{3} (2)^{3} (- 3)^{4} (4)^{2}

= \frac{9.8}{2} \times \frac{7.6.5}{6} .8.81.16 = 36 \times 35 \times 8 \times 81 \times 16

= 13063680

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