Write down all the ten digit numbers whose digital sum is 2. (The digital sum of a number is the sum of the digits of the number. The digital sum of 4022 is $$4+0+2+2=8$$.) Find the sum of all the 10 digit numbers with digital sum 2.

A 10-digit number with digital sum 2 can either use a single 2 as one digit with all other digits 0, giving one such number 2000000000, or use two 1s with all other digits 0. Since the leading digit can't be 0, listing all valid placements of the 2 and the pairs of 1s gives 10 such numbers in total. Adding all 10 numbers together (each contributing its place value once) gives the total sum $$11111111111$$.