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Nov 2017 p63 q6
2774
A car park has spaces for 18 cars, arranged in a line. On one day there are 5 cars, of different makes, parked in randomly chosen positions and 13 empty spaces.
Find the number of possible arrangements of the 5 cars in the car park.
Find the probability that the 5 cars are not all next to each other.
Solution
(i) The number of possible arrangements of the 5 cars in the car park is given by the permutation formula:
(ii) To find the probability that the 5 cars are not all next to each other, first calculate the probability that they are all next to each other. Consider the 5 cars as a single block, which can be arranged in \(5!\) ways. There are 14 possible positions for this block in the line of 18 spaces:
\(5! \times 14 = 1,680\)
The probability that the cars are all next to each other is:
\(\frac{1,680}{1,028,160} = 0.001634\)
Therefore, the probability that the cars are not all next to each other is:
