How many positive five digit numbers have a 1, 3, 7 or 5 in them?
We can use the principle of inclusion-exclusion to count the number of positive five-digit numbers that have a 1, 3, 7, or 5 in them.
First, we count the total number of five-digit numbers: there are 9 possible digits for the first digit (since it cannot be 0), and 10 possible digits for each of the remaining four digits. Therefore, there are $9\cdot10^4$ five-digit numbers.
Next, we count the number of five-digit numbers that do not have a 1, 3, 7, or 5 in them. For each digit, there are 5 possible digits remaining (since we cannot use 1, 3, 7, or 5). Therefore, there are $5^5$ such numbers.
Therefore, by the principle of inclusion-exclusion, the number of positive five-digit numbers that have a 1, 3, 7, or 5 in them is $9\cdot10^4 - 5^5 = 90,\!000 - 3,\!125 = \boxed{86,875}$.
To find the number of positive five-digit numbers that have a 1, 3, 7, or 5 in them, we can use the principle of complement.
Step 1: Find the total number of five-digit numbers.
To find the total number of five-digit numbers, we need to consider all the possible digits for each place value. For the first digit, it can be any digit from 1 to 9 (excluding 0). For the next four digits, they can be any digit from 0 to 9.
So, the total number of five-digit numbers is: 9 × 10 × 10 × 10 × 10 = 90,000.
Step 2: Find the number of five-digit numbers that do not have a 1, 3, 7, or 5.
To find the number of five-digit numbers that do not have a 1, 3, 7, or 5, we calculate the total number of possible numbers and subtract the number of numbers that do have a 1, 3, 7, or 5.
For each digit in the number, there are 6 possible digits that are not 1, 3, 7, or 5 (0, 2, 4, 6, 8, 9). Since we have 5 digit positions, the total number of five-digit numbers that do not have a 1, 3, 7, or 5 is: 6 × 6 × 6 × 6 × 6 = 7,776.
Step 3: Find the number of five-digit numbers that have a 1, 3, 7, or 5.
To find the number of five-digit numbers that have a 1, 3, 7, or 5, we subtract the number of five-digit numbers that do not have a 1, 3, 7, or 5 from the total number of five-digit numbers.
Number of five-digit numbers with a 1, 3, 7, or 5 = Total number of five-digit numbers - Number of five-digit numbers without a 1, 3, 7, or 5
Number of five-digit numbers with a 1, 3, 7, or 5 = 90,000 - 7,776 = 82,224.
Therefore, there are 82,224 positive five-digit numbers that have a 1, 3, 7, or 5 in them