Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 95888 by i jagooll last updated on 28/May/20

(1−2x)^5 (2+x)^6 = a+bx+cx^2 +dx^3 +... find : a,b,c and d

$$\left(\mathrm{1}−\mathrm{2x}\right)^{\mathrm{5}} \left(\mathrm{2}+\mathrm{x}\right)^{\mathrm{6}} =\:\mathrm{a}+\mathrm{bx}+\mathrm{cx}^{\mathrm{2}} +\mathrm{dx}^{\mathrm{3}} +... \\ $$$$\mathrm{find}\::\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{and}\:\mathrm{d}\: \\ $$

Answered by john santu last updated on 28/May/20

⇒(2+x) Σ_(k = 0) ^5 C(5,k) (2)^(5−k) (−(3x+2x^2 ))^k clearly a = 64 term containing x ⇒(2+x)C(5,1) 2^4 (−3x−2x^2 ) = 80(2+x)(−3x−2x^2 ) = −480x−560x^2 −160x^3 so b = −480 term containing x^2 = (2+x)C(5,2) 2^3 (3x+2x^2 )^2 = 80(2+x)(9x^2 +12x^3 +4x^4 ) so term x^2 = 560x^2 +1440x^2 then we get c = 2000 term containing x^3 = (2+x) C(5,3) 2^(2 ) (−3x−2x^2 )^3 = 40(2+x)(−27x^3 +...) = −2160x^3 +... so term x^3 = −160x^3 +1920x^3 +720x^3 −2160x^3 = 320x^3 then d = 320

$$\Rightarrow\left(\mathrm{2}+\mathrm{x}\right)\:\underset{\mathrm{k}\:=\:\mathrm{0}} {\overset{\mathrm{5}} {\sum}}\mathrm{C}\left(\mathrm{5},\mathrm{k}\right)\:\left(\mathrm{2}\right)^{\mathrm{5}−\mathrm{k}} \:\left(−\left(\mathrm{3x}+\mathrm{2x}^{\mathrm{2}} \right)\right)^{\mathrm{k}} \\ $$$$\mathrm{clearly}\:\mathrm{a}\:=\:\mathrm{64}\: \\ $$$$\mathrm{term}\:\mathrm{containing}\:\:\mathrm{x}\:\Rightarrow\left(\mathrm{2}+\mathrm{x}\right)\mathrm{C}\left(\mathrm{5},\mathrm{1}\right)\:\mathrm{2}^{\mathrm{4}} \:\left(−\mathrm{3x}−\mathrm{2x}^{\mathrm{2}} \right)\: \\ $$$$=\:\mathrm{80}\left(\mathrm{2}+\mathrm{x}\right)\left(−\mathrm{3x}−\mathrm{2x}^{\mathrm{2}} \right)\: \\ $$$$=\:−\mathrm{480x}−\mathrm{560x}^{\mathrm{2}} −\mathrm{160x}^{\mathrm{3}} \\ $$$$\mathrm{so}\:\mathrm{b}\:=\:−\mathrm{480} \\ $$$$\mathrm{term}\:\mathrm{containing}\:\mathrm{x}^{\mathrm{2}} \\ $$$$=\:\left(\mathrm{2}+\mathrm{x}\right)\mathrm{C}\left(\mathrm{5},\mathrm{2}\right)\:\mathrm{2}^{\mathrm{3}} \:\left(\mathrm{3x}+\mathrm{2x}^{\mathrm{2}} \right)^{\mathrm{2}} \\ $$$$=\:\mathrm{80}\left(\mathrm{2}+\mathrm{x}\right)\left(\mathrm{9x}^{\mathrm{2}} +\mathrm{12x}^{\mathrm{3}} +\mathrm{4x}^{\mathrm{4}} \right) \\ $$$$\mathrm{so}\:\mathrm{term}\:\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{560x}^{\mathrm{2}} +\mathrm{1440x}^{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{we}\:\mathrm{get}\:\mathrm{c}\:=\:\mathrm{2000} \\ $$$$\mathrm{term}\:\mathrm{containing}\:\mathrm{x}^{\mathrm{3}} \\ $$$$=\:\left(\mathrm{2}+\mathrm{x}\right)\:\mathrm{C}\left(\mathrm{5},\mathrm{3}\right)\:\mathrm{2}^{\mathrm{2}\:} \left(−\mathrm{3x}−\mathrm{2x}^{\mathrm{2}} \right)^{\mathrm{3}} \\ $$$$=\:\mathrm{40}\left(\mathrm{2}+\mathrm{x}\right)\left(−\mathrm{27x}^{\mathrm{3}} +...\right) \\ $$$$=\:−\mathrm{2160x}^{\mathrm{3}} +... \\ $$$$\mathrm{so}\:\mathrm{term}\:\mathrm{x}^{\mathrm{3}} \:=\:−\mathrm{160x}^{\mathrm{3}} +\mathrm{1920x}^{\mathrm{3}} +\mathrm{720x}^{\mathrm{3}} −\mathrm{2160x}^{\mathrm{3}} \\ $$$$=\:\mathrm{320x}^{\mathrm{3}} \\ $$$$\mathrm{then}\:\mathrm{d}\:=\:\mathrm{320} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com