How many integers less than $100$ have digits that add to $8$?

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Let \(n\) be a positive integer with \(n<10\).

2. How many positive integers less than \(100\) have digit sum equal to \(n\)?

How many numbers have a digit sum equal to \(1\)? How many numbers have a digit sum equal to \(2\)? To \(3\)?

Can we spot a pattern emerging? Could we try arranging our results into a table?

3. How many positive integers less than \(1000\) have digit sum equal to \(n\)?

Can we use our findings from (ii) to help us? Try picking \(8\) again - can we see a pattern?

4. How many positive integers between \(500\) and \(999\) have digit sum equal to \(8\)?

We know that the hundreds digit must be greater than or equal to \(5\). What does that tell us about the digits in the tens and units columns?

5. How many positive integers less than \(1000\) have digit sum equal to \(8\), and one digit at least \(5\)?

Maybe we can use our answer to (iv) to help us here?

6. What is the total of the digit sums of the integers from \(0\) to \(999\) inclusive?

How many times does each digit appear in each position?