How many 5-digit numbers can you make from the digits 0, 2, 4, 6 and 8, if you can use any digit any number of times?

Sure, here is the step by step solution:

Step 1: Understand the problem We are asked to find out how many 5-digit numbers can be made from the digits 0, 2, 4, 6, and 8. We can use any digit any number of times.

Step 2: Identify the number of choices for each digit The first digit can be any of the 4 non-zero digits (2, 4, 6, 8), so we have 4 choices. The remaining four digits can be any of the 5 digits (0, 2, 4, 6, 8), so we have 5 choices for each of these digits.

Step 3: Apply the counting principle The counting principle states that if there are n ways to do one thing, and m ways to do another, then there are n*m ways to do both. So, the total number of 5-digit numbers is 4 * 5 * 5 * 5 * 5 = 2500.

So, there are 2500 different 5-digit numbers that can be made from the digits 0, 2, 4, 6, and 8, using any digit any number of times.