Question

Seven 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 $7$ persons being seated is ${}^7{P}_7=7!$ ways

If two are to be seated next to each other, treat them as one unit – and this one unit with the remaining $5$ can be seated in $6!$ ways – and in each one of these $6!$ ways the two persons can be interchanged in $2$ ways.

$\therefore$ probability$=\frac{2.6!}{7!}=\frac{2.6!}{7.6!}=\frac{2}{7}$