Consider the sequence $\frac{{ }^{\mathrm{n}}{\mathrm{C}}_0}{1.2.3}, \frac{{ }^{\mathrm{n}}{\mathrm{C}}_1}{2.3.4}, \frac{{ }^{\mathrm{n}}{\mathrm{C}}_2}{3.4.5}, \dots \dots$ if $\mathrm{n}=50$ then greatest term is

Let ${\mathrm{T}}_{\mathrm{r}+1}>{\mathrm{T}}_{\mathrm{r}}\Rightarrow \frac{{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}{(\mathrm{r}+1)(\mathrm{r}+2)(\mathrm{r}+3)}\ge \frac{{ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}-1}}{\mathrm{r}(\mathrm{r}+1)(\mathrm{r}+2)}$

$\Rightarrow \frac{\mathrm{n}-\mathrm{r}+1}{\mathrm{r}}\ge \frac{\mathrm{r}+3}{\mathrm{r}} \Rightarrow \mathrm{n}-2>2\mathrm{r}\text{ for }\mathrm{n}=50, \mathrm{r}<24$

So, 24^th and 25^th terms of sequence are greatest.