Q-1-Coefficient-of-a-8-b-4-c-9-d-9-in-expansion-of-abc-abd-acd-bcd-10-Q-2-Coefficient-of-1-x-in-expansion-of-1-x-n-1-1-x-n-Q-3-If-x-m-occurs-in-expansion-of-x-1-x-2-2n

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Question Number 48705 by rahul 19 last updated on 27/Nov/18

$$Q.1\to$$$$Coefficientof{a}^8{b}^4{c}^9{d}^{9}inexpansion$$$$of{(abc+abd+acd+bcd)}^{10}=?$$$$$$$$Q.2\to$$$$Coefficientof\frac{1}{x}inexpansionof$$$${(1+x)}^n{(1+\frac{1}{x})}^n=?$$$$$$$$Q.3\to$$$$If{x}^{m}occursinexpansionof$$$${(x+\frac{1}{x^2})}^{2n},thenitscoefficient=?$$

Commented by rahul 19 last updated on 27/Nov/18

$$OnrequestofMeritguidesir,$$$$thesearesomeproblems\dots .$$$$Hopeufindtheminteresting!:)$$

Commented by maxmathsup by imad last updated on 27/Nov/18

$$3)wehave{(x+\frac{1}{x^2})}^{2n}=x^{2n}{(1+\frac{1}{x})}^{2n}=x^{2n}\sum _{k=0}^{2n}{C}_{2n}^k{x}^{-k}=\sum _{k=0}^n{C}_{2n}^k{x}^{2n-k}sothe$$$$coefficientis{\lambda }_m=C_{2n}^k/2n-k=m\Rightarrow k=2n-m\Rightarrow {\lambda }_m=C_{2n}^{2n-m}=C_{2n}^m$$$$=\frac{\left(2n\right)!}{m!(2n-m)!}$$$$$$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Nov/18

$$2){(1+x)}^n\times {(1+x)}^n\times x^{-n}$$$$x^{-n}{(1+x)}^{2n}$$$$let(r+1)thtermof{(1+x)}^{2n}contains{x}^{n-1}$$$$2n_{c_r}{x}^r$$$$so{x}^r=x^{n-1}$$$$sorequired{x}^{-n}\times 2n_{c_{n-1}}\times x^{n-1}$$$$=\frac{\left(2n\right)!}{(n-1)!(n+1)!}{x}^{-1}socoefficientis\frac{\left(2n\right)!}{(n-1)!(n+1)!}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Nov/18

$$3)let(r+1)thtermcontains{x}^m$$$$2n_{c_r}{\left(x\right)}^{2n-r}{\left(x^{-2}\right)}^r$$$$2n_{c_r}{\left(x\right)}^{2n-3r}$$$$2n-3r=m$$$$r=\frac{2n-m}{3}$$$$sotermis$$$$2n_{c_{\frac{2n-m}{3}}}{\left(x\right)}^m$$$$socoefficientis\frac{\left(2n\right)!}{\left(\frac{2n-m}{3}\right)!\left(\frac{4n+m}{3}\right)!}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Nov/18

$$1){\left\{abcd(\frac{1}{d}+\frac{1}{c}+\frac{1}{b}+\frac{1}{a})\right\}}^{10}$$$$={\left(abcd\right)}^{10}{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})}^{10}$$$$C{\left(abcd\right)}^{10}\times \frac{1}{a^2}\times \frac{1}{b^6}\times \frac{1}{c}\times \frac{1}{d}istherequired$$$$termwhichcontains{a}^8{b}^4{c}^9{d}^9$$$$nowvalueofCis=\frac{10!}{2!6!1!1!}=\frac{10!}{2!6!}$$$$$$

Commented by rahul 19 last updated on 28/Nov/18

thanks sir Ur all answers are absolutely correct!

Commented by tanmay.chaudhury50@gmail.com last updated on 28/Nov/18

$$thankyouRahul\dots howisyourpreparation$$$$forIITJEE$$

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