If $\mathrm{A}$ and $\mathrm{B}$ are among 20 persons who sit at random along a round table, then the…
If $\mathrm{A}$ and $\mathrm{B}$ are among 20 persons who sit at random along a round table, then the probability that there are any six persons between $A$ and $B$ is
$\frac{1}{2}$
$\frac{5}{16}$
$\frac{2}{19}$
$\frac{2}{81}$
Solution
The no. of ways in which 18 people (excluding $A$ and $B$ ) sit in a round table with 6 people between $\mathrm{A} \& \mathrm{~B}={ }^{18} \mathrm{C}_6 \times 2 ! \times(13-1)$ !
Thus, probability $=\frac{18_{\mathrm{C}_6} \times 2 ! \times(13-1) ! \times 6 !}{(20-1) !}=\frac{2}{19}$