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Question Number 151211 by EDWIN88 last updated on 19/Aug/21

Find the coefficient of x^9 from expression (1+x)(1+2x^2 )(1+3x^3 )(1+4x^4 )(1+5x^5 )...(1+10x^(10) )

$$\:{Find}\:{the}\:{coefficient}\:{of}\:{x}^{\mathrm{9}} \: \\ $$$${from}\:{expression}\: \\ $$$$\:\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{3}} \right)\left(\mathrm{1}+\mathrm{4}{x}^{\mathrm{4}} \right)\left(\mathrm{1}+\mathrm{5}{x}^{\mathrm{5}} \right)...\left(\mathrm{1}+\mathrm{10}{x}^{\mathrm{10}} \right) \\ $$

Answered by Olaf_Thorendsen last updated on 19/Aug/21

f(x) = Π_(n=1) ^(10) (1+nx^n ) • 9 = 9+0 : coeff = 9×1 = 9 • 9 = 8+1 : coeff = 8×1 = 8 • 9 = 7+2 : coeff = 7×2 = 14 • 9 = 6+3 = 6+2+1 : coeff = 6×3+6×2×1 = 30 • 9 = 5+4 = 5+3+1 coeff = 5×4+5×3×1 = 35 • 9 = 4+3+2 coeff = 4×3×2 = 24 coeff of x^9 = 9+8+14+30+35+24 = 120

$${f}\left({x}\right)\:=\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\prod}}\left(\mathrm{1}+{nx}^{{n}} \right) \\ $$$$ \\ $$$$\bullet\:\mathrm{9}\:=\:\mathrm{9}+\mathrm{0}\::\:\mathrm{coeff}\:=\:\mathrm{9}×\mathrm{1}\:=\:\mathrm{9} \\ $$$$\bullet\:\mathrm{9}\:=\:\mathrm{8}+\mathrm{1}\::\:\mathrm{coeff}\:=\:\mathrm{8}×\mathrm{1}\:=\:\mathrm{8} \\ $$$$\bullet\:\mathrm{9}\:=\:\mathrm{7}+\mathrm{2}\::\:\mathrm{coeff}\:=\:\mathrm{7}×\mathrm{2}\:=\:\mathrm{14} \\ $$$$\bullet\:\mathrm{9}\:=\:\mathrm{6}+\mathrm{3}\:=\:\mathrm{6}+\mathrm{2}+\mathrm{1}\:: \\ $$$$\mathrm{coeff}\:=\:\mathrm{6}×\mathrm{3}+\mathrm{6}×\mathrm{2}×\mathrm{1}\:=\:\mathrm{30} \\ $$$$\bullet\:\mathrm{9}\:=\:\mathrm{5}+\mathrm{4}\:=\:\mathrm{5}+\mathrm{3}+\mathrm{1} \\ $$$$\mathrm{coeff}\:=\:\mathrm{5}×\mathrm{4}+\mathrm{5}×\mathrm{3}×\mathrm{1}\:=\:\mathrm{35} \\ $$$$\bullet\:\mathrm{9}\:=\:\mathrm{4}+\mathrm{3}+\mathrm{2} \\ $$$$\mathrm{coeff}\:=\:\mathrm{4}×\mathrm{3}×\mathrm{2}\:=\:\mathrm{24} \\ $$$$ \\ $$$$\mathrm{coeff}\:\mathrm{of}\:{x}^{\mathrm{9}} \:=\:\mathrm{9}+\mathrm{8}+\mathrm{14}+\mathrm{30}+\mathrm{35}+\mathrm{24}\:=\:\mathrm{120} \\ $$

Commented by liberty last updated on 19/Aug/21

yes typo in part •9 = 7+2

$$\mathrm{yes}\:\mathrm{typo}\:\mathrm{in}\:\mathrm{part}\:\bullet\mathrm{9}\:=\:\mathrm{7}+\mathrm{2}\: \\ $$

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