SOLUTION: Find the number of positive integers less than 1000 that contains at least one odd digit and at least one even digit.

To contain at least one odd digit and at least one even digit, 
and be less than 1000, a counting integer must have two or three 
digits.

Two digit counting numbers:

There are 90 2-digit numbers altogether, because there are 
9 ways to choose the first digit and 10 ways to choose the 
second digit, and 9×10 = 90

From that 90 we must subtract the number of two-digit numbers
with two even digits. There are 4 ways to choose the first
even digit and 5 ways to choose the second even digit, That's
4×5 or 20 two-digit numbers with both even digits we must subtract
from the 90.  That makes 90-20 or 70

From that 70 we must subtract the number of two-digit numbers
with two odd digits. There are 6 ways to choose the first
odd digit and 5 ways to choose the second odd digit, That's
5×5 or 25 two-digit numbers with both odd digits we must subtract
from the 70.  That makes 70-25 or 45.

So there are 45 2-digit numbers that have one even and one odd digit.

   
Three digit counting numbers:

There are 900 three-digit numbers altogether, because there are 9 
ways to choose the first digit, 10 ways to choose the second digit, 
and 10 ways to choose the third digit. 
That makes 9×10×10 = 900

From that 900 we must subtract the number of three-digit numbers
with all even digits. There are 4 ways to choose the first even 
digit, 5 ways to choose the second even digit, and 5 ways to
choose the third even digit. That's 4×5×5 or 100 three-digit 
numbers with all even digits which we must subtract from the 900.
That makes 900-100 or 800.

From that 800 we must subtract the number of three-digit numbers
with all odd digits. There are 5 ways to choose the first odd 
digit, 5 ways to choose the second odd digit, and 5 ways to
choose the third odd digit. That's 5×5×5 or 125 three-digit 
numbers with all odd digits which we must subtract from the 800.
That makes 800-125 or 675.

So there are 675 3-digit numbers that contain at least one odd 
digit and at least one even digit.

So 45 two-digit numbers plus 675 three-digit numbers = 720. 

Answer: 720.

Edwin