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Question
A typical PIN (personal identification number) is a sequence of any four symbols chosen from the 26 letters in the alphabet and the ten digits. If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol?
Solution
A PIN is a sequence of four symbols selected from 36 (26 letters + 10 digits) symbols.
By the fundamental principle of counting
There are 36 × 36 × 36 × 36 = 364 = 1,679,616 PINs in all.
When repetition is not allowed the multiplication rule can be applied to conclude that there are 36 × 35 × 34 × 33 = 1,413,720 different PINs
The number of PINs that contain at least one repeated symbol = 1,679,616 – 1,413,720 = 2,65,896
Thus, the probability that a randomly chosen PIN contains a repeated symbol is
`(265, 896)/(1, 679, 616)` = 1.583
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