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Question Number 115742 by bemath last updated on 28/Sep/20

what the cooefficient of x^(10) from (1+x)×(1+2x^2 )×(1+3x^3 )×...×(1+2020x^(2020) )

$${what}\:{the}\:{cooefficient}\:{of}\:{x}^{\mathrm{10}} \:{from} \\ $$$$\left(\mathrm{1}+{x}\right)×\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)×\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{3}} \right)×...×\left(\mathrm{1}+\mathrm{2020}{x}^{\mathrm{2020}} \right) \\ $$

Answered by mr W last updated on 28/Sep/20

x^(10) : 10=10 x^(9+1) ,x^(8+2) ,x^(7+3) ,x^(6+4) : 9×1+8×2+7×3+6×4=70 x^(7+2+1) ,x^(6+3+1) ,x^(5+4+1) ,x^(5+3+2) : 7×2×1+6×3×1+5×4×1+5×3×2=82 x^(4+3+2+1) : 4×3×2×1=24 Σ=10+95+72+24=186 ⇒coef. of x^(10) is 186.

$${x}^{\mathrm{10}} :\:\mathrm{10}=\mathrm{10} \\ $$$${x}^{\mathrm{9}+\mathrm{1}} ,{x}^{\mathrm{8}+\mathrm{2}} ,{x}^{\mathrm{7}+\mathrm{3}} ,{x}^{\mathrm{6}+\mathrm{4}} :\:\mathrm{9}×\mathrm{1}+\mathrm{8}×\mathrm{2}+\mathrm{7}×\mathrm{3}+\mathrm{6}×\mathrm{4}=\mathrm{70} \\ $$$${x}^{\mathrm{7}+\mathrm{2}+\mathrm{1}} ,{x}^{\mathrm{6}+\mathrm{3}+\mathrm{1}} ,{x}^{\mathrm{5}+\mathrm{4}+\mathrm{1}} ,{x}^{\mathrm{5}+\mathrm{3}+\mathrm{2}} :\:\mathrm{7}×\mathrm{2}×\mathrm{1}+\mathrm{6}×\mathrm{3}×\mathrm{1}+\mathrm{5}×\mathrm{4}×\mathrm{1}+\mathrm{5}×\mathrm{3}×\mathrm{2}=\mathrm{82} \\ $$$${x}^{\mathrm{4}+\mathrm{3}+\mathrm{2}+\mathrm{1}} :\:\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}=\mathrm{24} \\ $$$$\Sigma=\mathrm{10}+\mathrm{95}+\mathrm{72}+\mathrm{24}=\mathrm{186} \\ $$$$\Rightarrow{coef}.\:{of}\:{x}^{\mathrm{10}} \:{is}\:\mathrm{186}. \\ $$

Commented by mr W last updated on 28/Sep/20

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