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Question

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


Solution

$$\Rightarrow $$(1+2a)^4$$
$$\Rightarrow ^{4}C_1+{^{4}C_1}(2a)+{^{4}C_2}(2a)^2+{^{4}C_3}(2a)^3+{^{4}C_4}(2a)^4$$
$$\Rightarrow 1+8a+24a^2+32a^3+16a^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}2^2a^3+{^{5}C_4}2a^4-{^{5}C_5}a^5$$
$$\Rightarrow 32-80a+80a^3+10a^4-a^5$$
$$\therefore$$ Coefficient of $$a^4$$ is $$\Rightarrow 1\times 10a^4+(8a)(80a^3)+32a^3(-80a)+16a^4(32)$$
$$\Rightarrow 10a^4+640a^4-2560a^4+512a^4$$
$$\Rightarrow -1.398x^4$$.

1043068_1040863_ans_146f97e9c0a64606acb8cb58bcf6e0f0.jpg

Mathematics

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