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

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Solution

$(1 + 2 a)^{4}$ ..(1)
$(2 - a)^{5}$ ..(2)
Coefficient of $a^{4}$ term can be obtained when term $a$ from(1) and $a^{4}$ from(2) are multiplied.so combining all these coefficints we can find the coefficint of $a^{4}$ in given expansion,thus we get,
coeeficient of $a^{4}$ will be (taking 1 and 2 in equation)
$(a_{1}^{0} \times a_{2}^{4}) + (a_{1}^{1} \times a_{2}^{3}) + (a_{1}^{2} \times a_{2}^{2}) + (a_{1}^{3} \times a_{2}^{1}) + (a_{1}^{4} \times a_{2}^{0})$
$[^{4} C_{0} \times (1)^{4} (2 a)^{2} \times^{5} C_{4} \times 2^{1} \times (- a)^{5 - 1}]$ + $[^{4} C_{1} \times (1)^{3} (2 a)^{1} \times^{5} C_{3} \times 2^{2} \times (- a)^{3}]$ + $[^{4} C_{2} \times (1)^{2} (2 a)^{2} \times^{5} C_{2} \times 2^{3} \times (- a)^{2}]$ + $[^{4} C_{3} \times (1)^{1} (2 a)^{3} \times^{5} C_{1} \times 2^{4} \times (- a)^{1}]$ + $[^{4} C_{4} \times (1)^{0} (2 a)^{4} \times^{5} C_{0} \times 2^{5} \times (- a)^{0}]$
$10 a^{4} - 320 a^{4} + 1920 a^{4} - 2560 a^{4} + 512 a^{4}$
$- 438 a^{4}$
Therefore coeffcient of $a^{4}$ is $- 438$

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