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The first digit can be any one of 9 digits (anything but zero). For each of those ...

The second digit can be any one of 9 digits (anything but the previous one). For each of those ...

The third digit can be any one of 9 digits (anything but the previous one). For each of those ...

The fourth digit can be any one of 9 digits (anything but the previous one). For each of those ...

The fifth digit can be any one of 9 digits (anything but the previous one).

So the total number of possible arrangements is (9 x 9 x 9 x 9 x 9) = 59,049

out of a total of 99,999 different 5-digit numbers, or 59.05%.

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