# Find the coefficient of $x^5$ in the expansion of the product $(1+2x)^6(1-x)^7$

$\begin{array}{1 1}(A)\;181\\(B)\;161\\(C)\;151\\(D)\;171\end{array}$

Toolbox:
• $(a+b)^n=nC_0a^n+nC_1a^{n-1}b+nC_2a^{n-2}b^2+.........nC_ra^{n-r}b^r+......nC_nb^n$
$(1+2x)^6=6C_0.1+6C_1(2x)+6C_2(2x)^2+6C_2(2x)^3+6C_4(2x)^4+6C_5(2x)^5+6C_6(2x)^6$
$\Rightarrow 1+12x+60x^2+20\times 8x^3+15\times 16x^4+6\times 32x^6+64x^5$
$\Rightarrow 1+12x+60x^2+160x^3+240x^4+192x^5+64x^6$-----(1)
$(1-x)^7=1-7C_1x+7C_2x^2-7C_3x^3+7C_4x^4-7C_5x^5+....7C_6x^6-7C_7x^7$
$\Rightarrow 1-7x+21x^2-35x^3+35x^4-21x^5+7x^6-x^7$-------(2)
Multiplying (1) and (2) and collecting the coefficient of $x^5$
$\therefore$ Coefficient of $x^5$ in the product $(1+2x)^6(1-x)^7$
$\Rightarrow 1\times (-21)+12\times 35+60\times (-35)+160\times 21+240\times (-7)+192\times 1$
$\Rightarrow -21+420-2100+3360-1680+192$
$\Rightarrow 171$
Hence (D) is the correct answer.