1. How many integers are there between 0 and $10^{5}$ having the digit sum equal to 8 ?
1. Ans: 495

Each integer can be written as $\overline{x_{1} x_{2} x_{3} x_{4} x_{5}}$ where each $x_{t}=0,1,2, \ldots, 9$ with $x_{1}+x_{2}+$ $x_{3}+x_{4}+x_{5}=8$. The number of non-negative integer solutions to the above equation is 495 . So there are 495 such integers.