Question

Seven persons draw lots for the occupancy of the six seats in a first class railway compartments: find the chance $\left(1\right)$ that two specified persons obtain opposite seats, $\left(2\right)$ that they obtain adjacent seats on the same side.

Answer 1

Let the two persons be $A$ and $B$

Probability of getting a seat for let say $A$ $=\frac{6}{7}$

Now there is only one seat opposite to $A$

Now probablility that $B$ is oppsoite to $A$ $=\frac{6}{7}\times \frac{1}{6}=\frac{1}{7}$

$--------------------------------$

Any two adjacent seats can be selected in ${}^7{C}_2=21$ ways

Now $A$ and $B$ can be seated adjacent in $4$ ways

$\therefore$ required probability $=\frac{4}{21}$