Disclaimer: I'm assuming that you want any of the digit to be even, and the other to be odd, and not that you mean "the first digit odd and the second even".
You can note that there are 100 numbers with two digits, from 0 (which you can read as 00) to 99.
Now, we're only interested in those who have one odd digit and one even digit. Let's see if we can find a pattern in the numbers we do or don't want to include in our list:
Consider the numbers from 00 to 09. Since the first digit is zero, which is even, the unit digit must be odd. So, we must exlude the following numbers:
while the following numbers are ok:
Now, consider the numbers from 10 to 19. This time the first digit is one, which is odd, so the unit digit must be even. Now, we must exlude the following numbers:
while the following numbers are ok:
Can you see the pattern? If you consider any 10 consecutive numbers with the same first digit, you will always keep half of the numbers and discard the other half. In fact, if the first digit is even, you will keep
, while you will discard
.
If, instead, the first digit is odd, your choice will be the opposite: you will discard
, while you will keep
.
So, we've just proven that exactly half of the numbers with two digits have one odd digit and one even digit. We observed at the beginning that there are 100 two-digits number in total, so there are 50 numbers with one even digit and one odd digit.