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Question Number 199054 by cortano12 last updated on 27/Oct/23
x
$$\:\:\boldsymbol{{x}} \\ $$
Answered by AST last updated on 27/Oct/23
(1+x)(2+x^2 )(3+x^3 )(4+x^4 )(5+x^5 )(6+x^6 ) 8=6+2=5+3=5+2+1=4+3+1 6!((1/(12))+(1/(15))+(1/(10))+(1/(12)))=240
$$\left(\mathrm{1}+{x}\right)\left(\mathrm{2}+{x}^{\mathrm{2}} \right)\left(\mathrm{3}+{x}^{\mathrm{3}} \right)\left(\mathrm{4}+{x}^{\mathrm{4}} \right)\left(\mathrm{5}+{x}^{\mathrm{5}} \right)\left(\mathrm{6}+{x}^{\mathrm{6}} \right) \\ $$$$\mathrm{8}=\mathrm{6}+\mathrm{2}=\mathrm{5}+\mathrm{3}=\mathrm{5}+\mathrm{2}+\mathrm{1}=\mathrm{4}+\mathrm{3}+\mathrm{1} \\ $$$$\mathrm{6}!\left(\frac{\mathrm{1}}{\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{15}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{12}}\right)=\mathrm{240} \\ $$
Answered by mr W last updated on 27/Oct/23
(1+x)(2+x^2 )(3+x^3 )...(6+x^6 ) 8=2+6=3+5=1+2+5=1+3+4 ⇒coef. of x^8 is 1×3×4×5+1×2×4×6+ 3×4×6+2×5×6 =240
$$\left(\mathrm{1}+{x}\right)\left(\mathrm{2}+{x}^{\mathrm{2}} \right)\left(\mathrm{3}+{x}^{\mathrm{3}} \right)…\left(\mathrm{6}+{x}^{\mathrm{6}} \right) \\ $$$$\mathrm{8}=\mathrm{2}+\mathrm{6}=\mathrm{3}+\mathrm{5}=\mathrm{1}+\mathrm{2}+\mathrm{5}=\mathrm{1}+\mathrm{3}+\mathrm{4} \\ $$$$\Rightarrow{coef}.\:{of}\:{x}^{\mathrm{8}} \:{is}\: \\ $$$$\mathrm{1}×\mathrm{3}×\mathrm{4}×\mathrm{5}+\mathrm{1}×\mathrm{2}×\mathrm{4}×\mathrm{6}+ \\ $$$$\mathrm{3}×\mathrm{4}×\mathrm{6}+\mathrm{2}×\mathrm{5}×\mathrm{6} \\ $$$$=\mathrm{240} \\ $$

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