Question Number 82658 by jagoll last updated on 23/Feb/20
$${coefficient}\:{x}^{\mathrm{6}} \:{from}\:{expressi}\: \\ $$$$\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{6}\:} ×\:\left({x}^{\mathrm{2}} +{x}+\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} \:? \\ $$
Commented by mr W last updated on 23/Feb/20
$$=\mathrm{2}^{\mathrm{6}} \left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{6}} \left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{8}} \\ $$$$=\mathrm{2}^{\mathrm{6}} \left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{14}} \\ $$$$=\mathrm{2}^{\mathrm{6}} \Sigma\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{14}−{k}} }{C}_{{k}} ^{\mathrm{14}} {x}^{{k}} \\ $$$${k}=\mathrm{6}: \\ $$$$\mathrm{2}^{\mathrm{6}} ×\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{14}−\mathrm{6}} }×{C}_{\mathrm{6}} ^{\mathrm{14}} =\frac{{C}_{\mathrm{6}} ^{\mathrm{14}} }{\mathrm{2}^{\mathrm{2}} }=\frac{\mathrm{3003}}{\mathrm{4}} \\ $$
Commented by jagoll last updated on 23/Feb/20
$${thank}\:{you}\:{mister} \\ $$
Answered by TANMAY PANACEA last updated on 23/Feb/20
$$\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{6}} ×\left(\frac{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} \\ $$$$\frac{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{6}} }{\mathrm{4}^{\mathrm{4}} }×\left\{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} \right\}^{\mathrm{4}} \\ $$$$\frac{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{14}} }{\mathrm{4}^{\mathrm{4}} } \\ $$$${now}\:{let}\:{r}+\mathrm{1}\:{th}\:{term}\:{contains}\:{x}^{\mathrm{6}} \\ $$$$\frac{\mathrm{14}_{{C}_{{r}} } }{\mathrm{4}^{\mathrm{4}} }×\left(\mathrm{2}{x}\right)^{\mathrm{14}−{r}} ×\mathrm{1}^{{r}} \\ $$$${so}\:{x}^{\mathrm{14}−{r}} ={x}^{\mathrm{6}} \rightarrow{r}=\mathrm{8} \\ $$$$\mathrm{14}{c}_{\mathrm{8}} ×\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{4}} }×\mathrm{2}^{\mathrm{6}} ×{x}^{\mathrm{6}} \\ $$
Commented by jagoll last updated on 23/Feb/20
$$\frac{\mathrm{14}×\mathrm{13}×\mathrm{12}×\mathrm{11}×\mathrm{9}×\mathrm{8}}{\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}}\:×\:\mathrm{64}\: \\ $$