Even three-digit codes with a repeated digit

Question

A keypad lock uses a string of three digits (leading zeros allowed). The code is known to be even, none of its digits is a $6$ , and at least one digit occurs more than once. How many codes are consistent with this information?

Short answer

100. Without the repeat condition: the last digit is one of $0,2,4,8$ and the other two are any of the nine digits other than $6$ , giving $9\cdot9\cdot4=324$ . Codes with three distinct digits number $4\cdot8\cdot7=224$ , so $324-224=100$ codes repeat a digit.

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Lessons behind this question

Counting and Combinatorics

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