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Question Number 63934 by gunawan last updated on 11/Jul/19
The coefficient of x^5 in the expansion of (2−x+3x^2 )^6 is
$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:\:{x}^{\mathrm{5}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{2}−{x}+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{6}} \:\:\mathrm{is} \\ $$
Commented by Prithwish sen last updated on 14/Jul/19
((6!)/(α!β!γ!)) . 2^α .(−1)^β .3^γ where α+β+γ = 6 and β +2γ = 5 𝛂 𝛃 𝛄 3 1 2 2 3 1 1 5 0 ∴ the coefficient is ((6!)/(3!1!2!))2^3 (−1)^1 3^2 +((6!)/(2!3!1!))2^2 (−1)^3 3^1 +((6!)/(1!5!0!))2^1 (−1)^5 3^0 =−4320−720−12=−5052. please check.
$$\frac{\mathrm{6}!}{\alpha!\beta!\gamma!}\:.\:\mathrm{2}^{\alpha} .\left(−\mathrm{1}\right)^{\beta} .\mathrm{3}^{\gamma} \\ $$$$\mathrm{where}\:\alpha+\beta+\gamma\:=\:\mathrm{6}\:\mathrm{and}\:\beta\:+\mathrm{2}\gamma\:=\:\mathrm{5} \\ $$$$\:\:\:\:\:\boldsymbol{\alpha}\:\:\:\boldsymbol{\beta}\:\:\:\boldsymbol{\gamma} \\ $$$$\:\:\:\:\:\mathrm{3}\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\mathrm{2}\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\mathrm{1}\:\:\:\mathrm{5}\:\:\:\:\:\:\:\mathrm{0} \\ $$$$\therefore\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{is} \\ $$$$\frac{\mathrm{6}!}{\mathrm{3}!\mathrm{1}!\mathrm{2}!}\mathrm{2}^{\mathrm{3}} \left(−\mathrm{1}\right)^{\mathrm{1}} \mathrm{3}^{\mathrm{2}} +\frac{\mathrm{6}!}{\mathrm{2}!\mathrm{3}!\mathrm{1}!}\mathrm{2}^{\mathrm{2}} \left(−\mathrm{1}\right)^{\mathrm{3}} \mathrm{3}^{\mathrm{1}} +\frac{\mathrm{6}!}{\mathrm{1}!\mathrm{5}!\mathrm{0}!}\mathrm{2}^{\mathrm{1}} \left(−\mathrm{1}\right)^{\mathrm{5}} \mathrm{3}^{\mathrm{0}} \\ $$$$=−\mathrm{4320}−\mathrm{720}−\mathrm{12}=−\mathrm{5052}.\:\mathrm{please}\:\mathrm{check}. \\ $$
Answered by ajfour last updated on 14/Jul/19
[3x^2 +(x−2)]^6 =(3x^2 )^6 +6(3x^2 )^5 (x−2) +15(3x^2 )^4 (x−2)^2 +20(3x^2 )^3 (x−2)^3 + 15(3x^2 )^2 (x−2)^4 +6(3x^2 )(x−2)^5 +(x−2)^6 C_5 =15×9×(−32) 6×3×40−12 C_5 =−270×16+45×16−12 coeff. of x^5 = −(225×16+12) = −3612.
$$\left[\mathrm{3}{x}^{\mathrm{2}} +\left({x}−\mathrm{2}\right)\right]^{\mathrm{6}} =\left(\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{6}} +\mathrm{6}\left(\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{5}} \left({x}−\mathrm{2}\right) \\ $$$$\:\:\:+\mathrm{15}\left(\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{4}} \left({x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{20}\left(\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{3}} \left({x}−\mathrm{2}\right)^{\mathrm{3}} \\ $$$$\:+\:\mathrm{15}\left(\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} \left({x}−\mathrm{2}\right)^{\mathrm{4}} +\mathrm{6}\left(\mathrm{3}{x}^{\mathrm{2}} \right)\left({x}−\mathrm{2}\right)^{\mathrm{5}} \\ $$$$\:\:+\left({x}−\mathrm{2}\right)^{\mathrm{6}} \\ $$$${C}_{\mathrm{5}} =\mathrm{15}×\mathrm{9}×\left(−\mathrm{32}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{6}×\mathrm{3}×\mathrm{40}−\mathrm{12} \\ $$$${C}_{\mathrm{5}} =−\mathrm{270}×\mathrm{16}+\mathrm{45}×\mathrm{16}−\mathrm{12} \\ $$$${coeff}.\:{of}\:{x}^{\mathrm{5}} \:=\:−\left(\mathrm{225}×\mathrm{16}+\mathrm{12}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:−\mathrm{3612}. \\ $$

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