Question

Nine persons are to be seated on one side of a straight table. What is the probability that two particular persons will be seated next to each other?

Answer 1

Total number of ways of $9$ persons being seated is ways ${}^9{P}_9=9!$ ways

If two are to be seated next to each other, treat them as one unit – and this one unit with the remaining $7$ can be seated in $8!$ ways– and in each one of these $8!$ ways the two persons can be interchanged in $2$ ways.

$\therefore$probability $=\frac{2\times 8!}{9!}=\frac{2}{9}$