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

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Solution

The binomial coefficient of $(a x)^{2} \times (- b y)^{3} \times (c z)^{4} = (- a^{2} b^{3} c^{4}) \times (x^{2} y^{3} z^{4})$ is given by $\frac{9!}{2! \times 3! \times 4!} = 1260$
So, the numerical coefficient of $x^{2} y^{3} z^{4} = 24 \times - a^{2} b^{3} c^{4} = - 24 a^{2} b^{3} c^{4}$

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