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Question Number 58373 by Tawa1 last updated on 22/Apr/19

$$\mathrm{Find}\mathrm{the}\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^6\mathrm{in}{(2\mathrm{x}+1)}^6{({\mathrm{x}}^2+\mathrm{x}+\frac{1}{4})}^4$$

Commented by maxmathsup by imad last updated on 24/Apr/19

$$wehave{(2x+1)}^6{(x^2+x+\frac{1}{4})}^4=2^6{(x+\frac{1}{2})}^6{(x+\frac{1}{2})}^8$$$$=2^6{(x+\frac{1}{2})}^{14}=2^6\sum _{k=0}^{14}{C}_{14}^k{x}^k{\left(\frac{1}{2}\right)}^{14-k}=\frac{2^6}{2^{14}}\sum _{k=0}^{14}{C}_{14}^k{2}^k{x}^{k}sothecoefficient$$$$of{x}^{6}is{a}_6=2^{-8}{C}_{14}^6{2}^6=\frac{1}{4}{C}_{14}^6=\frac{1}{4}\frac{14!}{6!8!}.$$

Answered by tanmay last updated on 22/Apr/19

$${(2x+1)}^6{\left[{(x+\frac{1}{2})}^2\right]}^4$$$${(2x+1)}^6{(x+\frac{1}{2})}^8$$$$$$$$\mathit{o}\mathit{r}\mathit{m}\mathit{e}\mathit{t}\mathit{h}\mathit{o}\mathit{d}$$$${(2\mathit{x}+1)}^6{\left[{(\mathit{x}+\frac{1}{2})}^2\right]}^4$$$$=2^6{(\mathit{x}+\frac{1}{2})}^6{(\mathit{x}+\frac{1}{2})}^8$$$$=2^6{(\mathit{x}+\frac{1}{2})}^{14}$$$$=2^6\times 14{\mathit{c}}_8\times {\mathit{x}}^6\times {\left(\frac{1}{2}\right)}^8$$$$=\frac{14\times 13\times 12\times 11\times 10\times 9}{6\times 5\times 4\times 3\times 2\times 2^2}$$$$=\frac{14\times 13\times 11\times 9}{6\times 4}$$$$=\frac{7\times 13\times 11\times 3}{4}=\frac{3003}{4}$$

Commented by Tawa1 last updated on 23/Apr/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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