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Rastapopoulos had announced that it woul...

Rastapopoulos had announced that it would issue six-digit identity numbers to all his soldiers using numbers 0 through 9. If each number has to differ from every other number in at least two places. What is the highest population the island can have?

A

99999

B

100000

C

999999

D

None of these

Text Solution

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The answer to this question is `10^5`. Instead of finding a six digit number with two  different digits in any two numbers, find a 5 digit number with 1 different digit in any two  numbers (which will always have a different digit between 2 numbers) and just add the last  digit in the units place. This 5 digit number can be anything from 0 to 99999 and can be  done in `10^5` ways. The `6^(​th)`​ digit is obtained by adding the first 5 digits and taking the units  place of the sum.
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