Find the coefficient of $x^{16}$ in the expansion of ${(x^{2} - 2 x)}^{10}$ .

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Solution

We have ${(x^{2} - 2 x)}^{10} = x^{20} {(1 - \frac{2}{x})}^{10}$
and since $x^{20}$ multiplies every term in the expansion of ${(1 - \frac{2}{x})}^{10}$ , we have in this expansion to seek the coefficient of the term which contains $\frac{1}{x^{4}}$
Hence the required coefficient ${^{10} C}_{4} {(- 2)}^{4} = \frac{10.9.8.7}{1.2.3.4} \times 16 = 3360$

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