How many integers between 1 and 10^21 are such that the sum of their digits is 2?

GRE Quant Problem Solving
1

Capture

Can any one explain the process to solve it?

2

Combinatorics. Notice that if the sum is 2, the number must either be of the form

  • “00000002” - of which there are 21 ways since we are looking for all numbers up to 1021
  • “11000000000” - of which there are \frac{21!}{2! 19!} → 210 ways.

Sum them up: 210 + 21 = 231.