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 Question from Jessica: I need to find the total number of 3 digit codes using the digits 1 through 9. How many of these codes would be odd numbers, and how many even?

Hi Jessica,

Let's imagine that we are going to write down all the even ones. I want to start with the units digit. Since the number is to be even its units digit must be even and hence I have four choices, 2,4,6 and 8. Now extend each one digit code to a two digit code by choosing a tens digit. I have nine choices for a tens digit and hence I have $4 \times 9$ possible two digit codes. Finally extend each of the two digit codes to a three digit code by choosing the hundreds digit which I can do in nine ways and I thus have $4 \times 9 \times 9$ possible three digit codes each of which is even.