Question
A man parks his car among ' $n$ ' cars standing in a row. His car not being parked at an end. On his return he finds that exactly $m$ of the ' $n$ ' cars are still there. What is the probability that both the cars parked on two sides of his car have left.
Options
Solution
Sol. [Key Concept: A man parks among $n$ cars: probability the neighbouring spots are empty.]
Clearly his car is at one of the crosses
$$| \times \times \times \times \ldots\ldots \times |$$
The number of ways in which the remaining ( $m - 1$ ) cars can take their places (excluding the car of the man) $= \ ^{n - 1}C_{m - 1}$
The number of ways in which the remaining ( $m - 1$ ) cars can take places leaving the two places on two sides of his car $= \ ^{n - 3}C_{m - 1}$
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