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Question
A car is parked among N cars standing in a row, but at either end. On his return, the owner finds that exactly r of the N places are still occupied. The probability that both the places neighboring his car are empty is
A car is parked among N cars standing in a row, but at either end. On his return, the owner finds that exactly r of the N places are still occupied. The probability that both the places neighboring his car are empty is
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Solution
The correct option is B N−3Cr−1N−1Cr−1
Finding r cars in N places means there are r−1 cars other that his own in N−1 places.
The number of way this is possible is N−1Cr−1.
But if the neighboring slots are empty, the r−1 cars must be parked in N−3 places,
which can be done in N−3Cr−1 ways.
Therefore the probability of the required event is N−3Cr−1N−1Cr−1
Finding r cars in N places means there are r−1 cars other that his own in N−1 places.
The number of way this is possible is N−1Cr−1.
But if the neighboring slots are empty, the r−1 cars must be parked in N−3 places,
which can be done in N−3Cr−1 ways.
Therefore the probability of the required event is N−3Cr−1N−1Cr−1
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