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Question
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
Solution
\[(1 + 2a )^4 (2 - a )^5 \]
\[ = [ ^{4}{}{C}_0 (2a )^0 + ^{4}{}{C}_1 (2a )^1 +^{4}{}{C}_2 (2a )^2 + ^{4}{}{C}_3 (2a )^3 +^{4}{}{C}_4 (2a )^4 ] \times \]
\[ [ ^{5}{}{C}_0 (2 )^5 ( - a )^0 +^{5}{}{C}_1 (2 )^4 ( - a )^1 + ^{5}{}{C}_2 (2 )^3 ( - a )^2 + ^{5}{}{C}_3 (2 )^2 ( - a )^3 + ^{5}{}{C}_4 (2 )^1 ( - a )^4 + ^{5}{}{C}_5 (2 )^0 ( - a )^5 ]\]
\[ = [1 + 8a + 24 a^2 + 32 a^3 + 16 a^4 ] \times [32 - 80a + 80 a^2 - 40 a^3 + 10 a^4 - a^5 ]\]
\[\text{ Coefficient of } a^4 = 10 - 320 + 1920 - 2560 + 512 = - 438\]
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