We define the digit sum of a non-negative integer to be the sum of its digits. For example, the digit sum of 123 is 1 + 2 + 3 = 6. Let n be a positive integer with n < 10. How many positive integers less than 1000 have digit sum equal to n?

First we should consider how many positive integers less than 100 have digit sum equal to n. The answer should be relatively straight forward for this - it is n+1.For example, if n = 3 then we can have 03,12,21,30, so 4 numbers in total.We shall use this result in our question.
Then we need to consider the following observations:If the number ABC has digit sum equal to n, then the number BC would have digit sum less than n.If B+C is less than n, we can find a unique A such that A+B+C = nsince n<10, the unique A must be a single-digit number.Combining the 3 facts together we can deduce that the number of 3-DIGIT numbers with digit sum n = 1+2....+n (i.e the sum of 2-digit numbers with digit sum less than n) = (n+1)n/2so finally we need to add the number of 2-or-1-digit numbers with digit sum n to the formula above, so the answer is(n+1)n/2 + (n+1) = (n+1)(n+2)/2