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Find the coefficient of x^4 in the expan...

Find the coefficient of `x^4` in the expansion of `(2-x+3x^2)^6dot`

Text Solution

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`(2-x+3x^(2))^(6) = [2-x(1-3x)]^(6)`
`= 2^(6) -.^(6)C_(1) 2^(5)x(1-3x) + .^(6)C_(2)2^(4)x^(2)`
`xx (1-3x)^(2) - .^(6)C_(3)2^(3)x^(3)(1-3x)^(3)`
` + .^(6)C_(4)2^(2)x^(4)(1-3x)^(4)-.^(6)C_(5)2x^(5)(1-3x)^(5)+.^(6)C_(6)2^(6)`
Obviously, `x^(4)` occurs in `3^(rd), 4^(th)` and `5^(th)` terms. Now, `3^(rd)` term is `15 xx 16x^(2)(1-6x+9x^(2))`. Here, the coefficient of `x^(4)` is `15 xx 16 xx 9 = 2160`. The `4^(th)` term is `-20 xx 8x^(3)(1-9x+27x^(2)-27x^(3))`. Here, the coefficient of `x^(4)` is `20 xx 8 xx 9 = 1440`. The `5^(th)` term is `15 xx 4x^(4) [ 1-4 xx 3 x + "....." + (3x)^(4)]`. Here, the coefficient of `x^(4)` is `15 xx 4 = 60`.
Hence, the coefficient of `x^(4)` is `2160 + 1440 + 60 = 3660`.
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