Question

There are 7 seats in a row. Three persons take seats at random. The probability that the middle seat is always occupied and no two persons are consecutive is

• A. $\frac{9}{70}$
• B. $\frac{9}{35}$
• C. $\frac{4}{35}$
• D. none of these

Answer 1

The correct option is C $\frac{4}{35}$

No. of possible outcomes = $\left(\begin{array}{c}7\\ 3\end{array}\right)\times 3!$
No. of favorable outcomes = $1\times \left(\begin{array}{c}2\\ 1\end{array}\right)\times \left(\begin{array}{c}2\\ 1\end{array}\right)\times 3!$
Hence probability $P=$$\frac{1\times \left(\begin{array}{c}2\\ 1\end{array}\right)\times \left(\begin{array}{c}2\\ 1\end{array}\right)\times 3!}{\left(\begin{array}{c}7\\ 3\end{array}\right)\times 3!}=\frac{4}{35}$