Problem:
How many three-digit positive integers have an odd number of even digits?
Answer Choices:
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Solution:
There are choices for the hundreds and tens digits of a three-digit positive integer. For each such choice, exactly 5 of the 10 possible choices for the units digit will make the total number of even digits odd. Therefore the number of three-digit positive integers with an odd number of even digits is .
A three-digit positive integer that has an odd number of even digits must have either three even digits or one even digit. There are ways for all the digits to be even, because the leading digit cannot be 0 . If there is exactly one even digit, that digit can occur in either the hundreds, the tens, or the ones place. These cases, respectively, give additional possibilities. Hence the number of three-digit positive integers with an odd number of even digits is .
The problems on this page are the property of the MAA's American Mathematics Competitions