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 87175 by john santu last updated on 03/Apr/20

$$\mathrm{if}\mathrm{in}\mathrm{the}\mathrm{expansion}\mathrm{of}{(1+\mathrm{x})}^{\mathrm{n}}\mathrm{the}$$$$\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^9\mathrm{is}\mathrm{the}\mathrm{aritmetic}$$$$\mathrm{mean}\mathrm{of}\mathrm{the}\mathrm{coeficients}\mathrm{of}{\mathrm{x}}^8\mathrm{and}$$$${\mathrm{x}}^{10}.\mathrm{find}\mathrm{the}\mathrm{possible}\mathrm{value}$$$$\mathrm{of}\mathrm{n}\mathrm{where}\mathrm{n}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{integer}$$

Commented by jagoll last updated on 03/Apr/20

$$\mathrm{i}\mathrm{can}\mathrm{try}$$$$2{\mathrm{C}}_9^{\mathrm{n}}={\mathrm{C}}_8^{\mathrm{n}}+{\mathrm{C}}_{10}^{\mathrm{n}}$$$$2\times \frac{\mathrm{n}!}{9!(\mathrm{n}-9)!}=\frac{\mathrm{n}!}{8!(\mathrm{n}-8)!}+\frac{\mathrm{n}!}{10!(\mathrm{n}-10)!}$$$$\frac{2}{9(\mathrm{n}-9)}=\frac{1}{1}+\frac{1}{10.9.(\mathrm{n}-10)(\mathrm{n}-9)}$$$$\frac{20(\mathrm{n}-10)-1}{10.9.(\mathrm{n}-9)(\mathrm{n}-10)}=1$$$$20\mathrm{n}-201=90({\mathrm{n}}^2-19\mathrm{n}+90)$$$${90\mathrm{n}}^2-1690\mathrm{n}+8301=0$$

Commented by john santu last updated on 03/Apr/20

$$\mathrm{good}\mathrm{sir}$$

Answered by john santu last updated on 03/Apr/20

$${2\mathrm{C}}_9^{\mathrm{n}}={\mathrm{C}}_8^{\mathrm{n}}+{\mathrm{C}}_{10}^{\mathrm{n}}$$$${4\mathrm{C}}_9^{\mathrm{n}}=({\mathrm{C}}_8^{\mathrm{n}}+{\mathrm{C}}_9^{\mathrm{n}})+({\mathrm{C}}_9^{\mathrm{n}}+{\mathrm{C}}_{10}^{\mathrm{n}})$$$${4\mathrm{C}}_9^{\mathrm{n}}={\mathrm{C}}_9^{\mathrm{n}+1}+{\mathrm{C}}_{10}^{\mathrm{n}+1}={\mathrm{C}}_{10}^{\mathrm{n}+2}$$$$\frac{4\mathrm{n}!}{9!(\mathrm{n}-9)!}=\frac{(\mathrm{n}+2)!}{10!(\mathrm{n}-8)!}$$$$4=\frac{{\mathrm{n}}^2+3\mathrm{n}+2}{10(\mathrm{n}-8)}\Rightarrow {\mathrm{n}}^2-37\mathrm{n}+322=0$$$$(\mathrm{n}-14)(\mathrm{n}-23)=0$$$$\mathrm{n}=14\mathrm{or}\mathrm{n}=23$$$$$$

Commented by peter frank last updated on 03/Apr/20

$$good$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com