Find the coefficient of $x$ in the expansion of $(1 - 3 x + 7 x^{2}) {(1 - x)}^{16} .$

18

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- 18

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19

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- 48

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Solution

The correct option is D $- 48$
$(1 - 3 x + 7 x^{2}) (1 - x)^{16}$

$\Rightarrow (1 - 3 x + 7 x^{2}) [16 C_{0} x^{0} - 16 C_{1} x + 16 C_{2} x^{2} . . . . . + 16 C_{n - 1} (- 1)^{n - 1} x^{n - 1} + n C_{n} (- 1) 6 n x^{n}]$

$\Rightarrow (1 - 3 x + 7 x^{2}) [1 - 16 C_{1} x + 16 C_{2} x^{2} + 16 C_{3} x^{3} + 16 C_{4} x^{4} + 16 C_{5} x^{5} + . . . . . + 16 C_{6} x^{16}]$

$\Rightarrow - 3.16 C_{1}$

$\Rightarrow - 3 \times \frac{16!}{1! 15!}$ .

$\Rightarrow - 3 \times 16$

$\Rightarrow - 48$

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