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f-x-x-1-x-2-x-3-x-50-Find-coefficient-of-x-49-

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Question Number 29777 by Joel578 last updated on 12/Feb/18
f(x) = (x − 1)(x − 2)(x − 3)...(x − 50) Find coefficient of x^(49)
$${f}\left({x}\right)\:=\:\left({x}\:−\:\mathrm{1}\right)\left({x}\:−\:\mathrm{2}\right)\left({x}\:−\:\mathrm{3}\right)…\left({x}\:−\:\mathrm{50}\right) \\ $$$$\mathrm{Find}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{49}} \\ $$
Commented by mrW2 last updated on 12/Feb/18
see Q29805
$${see}\:{Q}\mathrm{29805} \\ $$
Commented by 803jaideep@gmail.com last updated on 12/Feb/18
i guess yesterday ques was upto x−60
$$\mathrm{i}\:\mathrm{guess}\:\mathrm{yesterday}\:\mathrm{ques}\:\mathrm{was}\:\mathrm{upto}\:\mathrm{x}−\mathrm{60} \\ $$
Commented by Joel578 last updated on 12/Feb/18
It was my mistake. It should be (x − 50)
$$\mathrm{It}\:\mathrm{was}\:\mathrm{my}\:\mathrm{mistake}.\:\mathrm{It}\:\mathrm{should}\:\mathrm{be}\:\left({x}\:−\:\mathrm{50}\right) \\ $$
Commented by mrW2 last updated on 12/Feb/18
the more challenging question is: find the coefficient of x^k with 0≤k≤50.
$${the}\:{more}\:{challenging}\:{question}\:{is}: \\ $$$${find}\:{the}\:{coefficient}\:{of}\:{x}^{{k}} \:{with}\:\mathrm{0}\leqslant{k}\leqslant\mathrm{50}. \\ $$
Commented by 803jaideep@gmail.com last updated on 12/Feb/18
thn plz take an example and make us learn about some mid value sir ..like 30 or 25
$$\mathrm{thn}\:\mathrm{plz}\:\mathrm{take}\:\mathrm{an}\:\mathrm{example}\:\mathrm{and}\:\mathrm{make} \\ $$$$\mathrm{us}\:\mathrm{learn}\:\mathrm{about}\:\mathrm{some}\:\mathrm{mid}\:\mathrm{value}\:\mathrm{sir} \\ $$$$..\mathrm{like}\:\mathrm{30}\:\mathrm{or}\:\mathrm{25} \\ $$
Answered by puneet1789 last updated on 12/Feb/18
Commented by Joel578 last updated on 12/Feb/18
thank you very much
$${thank}\:{you}\:{very}\:{much} \\ $$
Answered by mrW2 last updated on 13/Feb/18
see Q29805 here n=50 a_k =−k, 1≤k≤50 P=a_1 a_2 ...a_(50) =50! coefficient of x^(49) : C_(49) =Σ_1 ^(50) a_i =−(1+2+...+50)=−((50×51)/2)=−1275 coefficient of x^(30) : C_(30) =(P/(1×2×3...30))+(P/(1×3×4...31))+(P/(1×4×5...32))+...+(P/(1×22×23...50)) +(P/(2×3×4...31))+(P/(2×4×5...32))+(P/(2×5×6...33))+...+(P/(2×22×23...50)) +(P/(3×4×5...32))+(P/(3×5×6...33))+(P/(3×6×7...34))+...+(P/(3×22×23...50)) .... +(P/(20×21×22...49))+(P/(20×22×23...50)) +(P/(21×22×23...50)) =.....
$${see}\:{Q}\mathrm{29805} \\ $$$${here}\:{n}=\mathrm{50} \\ $$$${a}_{{k}} =−{k},\:\mathrm{1}\leqslant{k}\leqslant\mathrm{50} \\ $$$${P}={a}_{\mathrm{1}} {a}_{\mathrm{2}} …{a}_{\mathrm{50}} =\mathrm{50}! \\ $$$$ \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{49}} : \\ $$$${C}_{\mathrm{49}} =\underset{\mathrm{1}} {\overset{\mathrm{50}} {\sum}}{a}_{{i}} =−\left(\mathrm{1}+\mathrm{2}+…+\mathrm{50}\right)=−\frac{\mathrm{50}×\mathrm{51}}{\mathrm{2}}=−\mathrm{1275} \\ $$$$ \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{30}} : \\ $$$${C}_{\mathrm{30}} =\frac{{P}}{\mathrm{1}×\mathrm{2}×\mathrm{3}…\mathrm{30}}+\frac{{P}}{\mathrm{1}×\mathrm{3}×\mathrm{4}…\mathrm{31}}+\frac{{P}}{\mathrm{1}×\mathrm{4}×\mathrm{5}…\mathrm{32}}+…+\frac{{P}}{\mathrm{1}×\mathrm{22}×\mathrm{23}…\mathrm{50}} \\ $$$$+\frac{{P}}{\mathrm{2}×\mathrm{3}×\mathrm{4}…\mathrm{31}}+\frac{{P}}{\mathrm{2}×\mathrm{4}×\mathrm{5}…\mathrm{32}}+\frac{{P}}{\mathrm{2}×\mathrm{5}×\mathrm{6}…\mathrm{33}}+…+\frac{{P}}{\mathrm{2}×\mathrm{22}×\mathrm{23}…\mathrm{50}} \\ $$$$+\frac{{P}}{\mathrm{3}×\mathrm{4}×\mathrm{5}…\mathrm{32}}+\frac{{P}}{\mathrm{3}×\mathrm{5}×\mathrm{6}…\mathrm{33}}+\frac{{P}}{\mathrm{3}×\mathrm{6}×\mathrm{7}…\mathrm{34}}+…+\frac{{P}}{\mathrm{3}×\mathrm{22}×\mathrm{23}…\mathrm{50}} \\ $$$$…. \\ $$$$+\frac{{P}}{\mathrm{20}×\mathrm{21}×\mathrm{22}…\mathrm{49}}+\frac{{P}}{\mathrm{20}×\mathrm{22}×\mathrm{23}…\mathrm{50}} \\ $$$$+\frac{{P}}{\mathrm{21}×\mathrm{22}×\mathrm{23}…\mathrm{50}} \\ $$$$=….. \\ $$

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