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# If-and-are-the-coefficient-of-x-8-and-x-24-respectively-in-the-expansion-of-x-4-2-1-x-4-10-in-powers-of-x-then-is-equal-to-

Question Number 131248 by bramlexs22 last updated on 03/Feb/21
$${If}\:\alpha\:{and}\:\beta\:{are}\:{the}\:{coefficient}\: \\$$$${of}\:{x}^{\mathrm{8}} \:{and}\:{x}^{−\mathrm{24}} \:{respectively}\: \\$$$${in}\:{the}\:{expansion}\:{of}\:\left[\:{x}^{\mathrm{4}} +\mathrm{2}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:\right]^{\mathrm{10}} \\$$$${in}\:{powers}\:{of}\:{x}\:{then}\:\frac{\alpha}{\beta}\:{is}\:{equal}\:{to}\: \\$$
Answered by EDWIN88 last updated on 03/Feb/21
$$\:\left[\left({x}^{\mathrm{4}} +{x}^{−\mathrm{4}} \right)+\mathrm{2}\:\right]^{\mathrm{10}} =\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}\:\begin{pmatrix}{\mathrm{20}}\\{\:\:{k}}\end{pmatrix}\:\left({x}^{\mathrm{4}} +{x}^{−\mathrm{4}} \right)^{\mathrm{20}−{k}} \:\left(\mathrm{2}\right)^{{k}} \\$$$$\:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}\:\begin{pmatrix}{\mathrm{20}}\\{\:\:{k}}\end{pmatrix}\:\mathrm{2}^{{k}} \:\left[\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{20}−{k}} {\sum}}\:\begin{pmatrix}{\mathrm{20}−{k}}\\{\:\:\:\:\:{k}}\end{pmatrix}\left({x}^{\mathrm{4}} \right)^{\mathrm{20}−{k}} \:\left({x}^{−\mathrm{4}} \right)^{{k}} \:\right] \\$$
Answered by mr W last updated on 03/Feb/21
$$\left[{x}^{\mathrm{4}} +\mathrm{2}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\right]^{\mathrm{10}} \\$$$$=\frac{\mathrm{1}}{{x}^{\mathrm{40}} }\left[{x}^{\mathrm{8}} +\mathrm{2}{x}^{\mathrm{4}} +\mathrm{1}\right]^{\mathrm{10}} \\$$$$=\frac{\mathrm{1}}{{x}^{\mathrm{40}} }\left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{20}} \\$$$$=\frac{\mathrm{1}}{{x}^{\mathrm{40}} }\underset{{k}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}{C}_{{k}} ^{\mathrm{20}} {x}^{\mathrm{4}{k}} \\$$$$=\underset{{k}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}{C}_{{k}} ^{\mathrm{20}} {x}^{\mathrm{4}{k}−\mathrm{40}} \\$$$${term}\:{x}^{\mathrm{8}} :\:\mathrm{4}{k}−\mathrm{40}=\mathrm{8}\:\Rightarrow{k}=\mathrm{12}\:\Rightarrow\alpha={C}_{\mathrm{12}} ^{\mathrm{20}} \\$$$${term}\:{x}^{−\mathrm{24}} :\:\mathrm{4}{k}−\mathrm{40}=−\mathrm{24}\:\Rightarrow{k}=\mathrm{4}\:\Rightarrow\beta={C}_{\mathrm{4}} ^{\mathrm{20}} \\$$$$\frac{\alpha}{\beta}=\frac{{C}_{\mathrm{12}} ^{\mathrm{20}} }{{C}_{\mathrm{4}} ^{\mathrm{20}} }=\frac{\mathrm{20}!}{\mathrm{12}!\mathrm{8}!}×\frac{\mathrm{4}!×\mathrm{16}!}{\mathrm{20}!}=\frac{\mathrm{16}×\mathrm{15}×\mathrm{14}×\mathrm{13}}{\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}} \\$$$$=\mathrm{26} \\$$
Commented by EDWIN88 last updated on 03/Feb/21
$${nice} \\$$