Question

A car is parked by an owner amongst 25 cars in a row, not at either end. On his return he finds that exactly 15 places are still occupied. The probability that both neighbouring places are empty is

• A. $\frac{91}{276}$
• B. $\frac{15}{184}$
• C. $\frac{15}{92}$
• D. $\frac{1}{25}$

Answer 1

The correct option is C $\frac{15}{92}$

Since only $15$ places are occupied, $10$ cars have left.
Now, if both the neighboring places are to be empty, then $8$ of the remaining $22$ cars (excluding his own car) have left.
Required probability = $\frac{{}^{22}{C}_8}{{}^{24}{C}_{10}}=\frac{22!}{14!\ast 8!}\ast \frac{14!\ast 10!}{24!}=\frac{10\ast 9}{23\ast 24}=\frac{15}{92}$