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Question

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

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Solution

(1+2a)4 ..(1)
(2a)5..(2)
Coefficient of a4 term can be obtained when term a from(1) and a4 from(2) are multiplied.so combining all these coefficints we can find the coefficint of a4 in given expansion,thus we get,
coeeficient of a4 will be (taking 1 and 2 in equation)
(a01×a42)+(a11×a32)+(a21×a22)+(a31×a12)+(a41×a02)
[4C0×(1)4(2a)2×5C4×21×(a)51]+[4C1×(1)3(2a)1×5C3×22×(a)3]+[4C2×(1)2(2a)2×5C2×23×(a)2]+[4C3×(1)1(2a)3×5C1×24×(a)1]+[4C4×(1)0(2a)4×5C0×25×(a)0]
10a4320a4+1920a42560a4+512a4
438a4
Therefore coeffcient of a4 is 438

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