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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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