Question Number 63832 by gunawan last updated on 10/Jul/19
$$\mathrm{The}\:\mathrm{largest}\:\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{24}} \:\mathrm{is} \\ $$
Commented by kaivan.ahmadi last updated on 10/Jul/19
$${you}\:{are}\:{right} \\ $$$$\mathrm{13}{th}\:{term}\:{is}\:{max} \\ $$$${since}\:\mathrm{24}\:{is}\:{even}\:\frac{\mathrm{24}}{\mathrm{2}}+\mathrm{1}=\mathrm{13}\:{term}\:{has}\:{max}\:{coff} \\ $$
Commented by Tony Lin last updated on 10/Jul/19
$${C}_{\mathrm{12}} ^{\mathrm{24}} =\frac{\mathrm{24}!}{\mathrm{12}!\mathrm{12}!}=\mathrm{2704156} \\ $$
Commented by kaivan.ahmadi last updated on 10/Jul/19
$$\mathrm{12}{th}\:{term}\:{has}\:{the}\:{largest}\:{coefficient}\: \\ $$$$\begin{pmatrix}{\mathrm{24}}\\{\mathrm{11}}\end{pmatrix} \\ $$
Commented by Tony Lin last updated on 10/Jul/19
$${C}_{\mathrm{11}} ^{\mathrm{24}} =\mathrm{2496144}<{C}_{\mathrm{12}} ^{\mathrm{24}} =\mathrm{2704156} \\ $$
Commented by mr W last updated on 10/Jul/19
$${max}.\:{of}\:{C}_{{r}} ^{{n}} \:{is}\:{when}\:{r}\:{is}\:{as}\:{close}\:{to}\:\frac{{n}}{\mathrm{2}} \\ $$$${as}\:{possible}. \\ $$$${if}\:{n}=\mathrm{24}\:\Rightarrow{r}=\frac{\mathrm{24}}{\mathrm{2}}=\mathrm{12} \\ $$$${if}\:{n}=\mathrm{25}\:\Rightarrow{r}=\mathrm{12}\:{or}\:\mathrm{13} \\ $$