Find the coefficient of $a^{4}$ in the product $(1 + 2 a)^{4} (2 - a)^{5}$ using binomial theorem.

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Solution

$\Rightarrow$ (1+2a)^4$
$\Rightarrow^{4} C_{1} +^{4} C_{1} (2 a) +^{4} C_{2} (2 a)^{2} +^{4} C_{3} (2 a)^{3} +^{4} C_{4} (2 a)^{4}$
$\Rightarrow 1 + 8 a + 24 a^{2} + 32 a^{3} + 16 a^{4}$
$(2 - a)^{5}$
$\Rightarrow^{5} C_{0} 2^{5} -^{5} C_{1} 2^{4} (a) +^{5} C_{2} 2^{3} (a)^{2} -^{5} C_{3} 2^{2} a^{3} +^{5} C_{4} 2 a^{4} -^{5} C_{5} a^{5}$
$\Rightarrow 32 - 80 a + 80 a^{3} + 10 a^{4} - a^{5}$
$∴$ Coefficient of $a^{4}$ is $\Rightarrow 1 \times 10 a^{4} + (8 a) (80 a^{3}) + 32 a^{3} (- 80 a) + 16 a^{4} (32)$
$\Rightarrow 10 a^{4} + 640 a^{4} - 2560 a^{4} + 512 a^{4}$
$\Rightarrow - 1.398 x^{4}$ .

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