Find the coefficient of $x^{16}$ in expansion of $(x^{2} - 2 x)^{1} 0$

^{10} C_{4} (- 2)^{5}

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^{10} C_{4} (- 2)^{4}

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^{10} C_{6} (- 1)^{6} (2)^{4}

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^{10} C_{5} (- 2)^{5}

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Solution

The correct options are
B $^{10} C_{6} (- 1)^{6} (2)^{4}$
C $^{10} C_{4} (- 2)^{4}$
$T_{r + 1} =^{10} C_{r} x^{20 - 2 r} \times (- 2 x)^{r}$
$=^{10} C_{r} x^{(20 - r)} (- 2)^{r}$
For the $x^{(16)}$
$⟹ 20 - r = 16$
$r = 4$
So coefficient would be $^{10} C_{4} (- 2)^{4}$
As $^{10} C_{4}$ = $^{10} C_{6}$
so it can be also written as $^{10} C_{6} (- 1)^{6} (- 2)^{4}$

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