KVPY SX SB Exam 19-Nov-2017 Question Paper

Thirty two persons ${\mathit{X}}_1,{\mathit{X}}_2,.....,{\mathit{X}}_{32}$ are randomly seated around a circular table at equal intervals. Two persons ${\mathit{X}}_{\mathit{i}}$ and ${\mathit{X}}_{\mathit{j}}$ are said to be within earshot of each other if there are at most three persons between them on the minor arc joining ${\mathit{X}}_{\mathit{i}}$ and ${\mathit{X}}_{\mathit{j}}$ . The probability that ${\mathit{X}}_1$ and ${\mathit{X}}_2$ are within earshot of each other is,
$\left(\mathrm{Here}\left(\frac{\mathit{n}}{\mathit{r}}\right)=\frac{\mathit{n}!}{\left(\mathit{n}-\mathit{r}\right)!\mathit{r}!}\right)$