Find the coefficient of

Question:

Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.

Solution:

$(1+2 a)^{4}(2-a)^{5}$

$\begin{aligned}=&\left[{ }^{4} C_{0}(2 a)^{0}+{ }^{4} C_{1}(2 a)^{1}+{ }^{4} C_{2}(2 a)^{2}+{ }^{4} C_{3}(2 a)^{3}+{ }^{4} C_{4}(2 a)^{4}\right] \times \\ &\left[{ }^{5} C_{0}(2)^{5}(-a)^{0}+{ }^{5} C_{1}(2)^{4}(-a)^{1}+{ }^{5} C_{2}(2)^{3}(-a)^{2}+{ }^{5} C_{3}(2)^{2}(-a)^{3}+{ }^{5} C_{4}(2)^{1}(-a)^{4}+{ }^{5} C_{5}(2)^{0}(-a)^{5}\right] \end{aligned}$

$=\left[1+8 a+24 a^{2}+32 a^{3}+16 a^{4}\right] \times\left[32-80 a+80 a^{2}-40 a^{3}+10 a^{4}-a^{5}\right]$

Coefficient of $a^{4}=10-320+1920-2560+512=-438$

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