Coefficient of $x^{5}$ in the expansion of the product $(1 + 2 x)^{6} (1 - x)^{7}$ is:

115

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165

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171

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none of the above

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Solution

The correct option is C $171$
We have,
$(1 + 2 x)^{6} (1 - x)^{7} = [1 +^{6} C_{1} (2 x) +^{6} C_{2} (2 x)^{2} + . . . +^{6} C_{6} (2 x)^{6}] \times [1 -^{7} C_{1} x +^{7} C_{2} x^{2} -^{7} C_{3} x^{3} + . . . -^{7} C_{6} x^{7}]$

$(1 + 2 x)^{6} (1 - x)^{7} = (1 + 12 x + 60 x^{2} + 160 x^{3} + 240 x^{4} + 192 x^{5} + . .) \times (1 - 7 x + 21 x^{2} - 35 x^{3} + 35 x^{4} - 21 x^{5} + . . .)$

$∴$ Coefficient of $x^{5}$ in the product

$= 1 (- 21) + 12 (35) + 60 (- 35) + 160 (21) + 240 (- 7) + 192 (1)$
$= - 21 + 420 - 2100 + 3360 - 1680 + 192 = 171$

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