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In the expansion of $\left(1+3 x+2 x^2\right)^6$ the coefficient of $x^{11}$ is
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The correct answer is:
$576$
$\left(1+3 x+2 x^2\right)^6=[1+x(3+2 x)]^6$
$=1+{ }^6 C_1 x(3+2 x)+{ }^6 C_2 x^2(3+2 x)^2$ $+{ }^6 C_3 x^3(3+2 x)^3+{ }^6 C_4 x^4(3+2 x)^4$ $+{ }^6 C_5 x^5(3+2 x)^5+{ }^6 C_6 x^6(3+2 x)^6$
Only $x^{11}$ gets from ${ }^6 C_6 x^6(3+2 x)^6$
${ }^6 C_6 x^6(3+2 x)^6=x^6(3+2 x)^6$
$\therefore \text { Coefficient of } x^{11}={ }^6 C_5 3.2^5=576$
$=1+{ }^6 C_1 x(3+2 x)+{ }^6 C_2 x^2(3+2 x)^2$ $+{ }^6 C_3 x^3(3+2 x)^3+{ }^6 C_4 x^4(3+2 x)^4$ $+{ }^6 C_5 x^5(3+2 x)^5+{ }^6 C_6 x^6(3+2 x)^6$
Only $x^{11}$ gets from ${ }^6 C_6 x^6(3+2 x)^6$
${ }^6 C_6 x^6(3+2 x)^6=x^6(3+2 x)^6$
$\therefore \text { Coefficient of } x^{11}={ }^6 C_5 3.2^5=576$
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