jmc

Let the three digit integer be $abc.$ We must have $a+b+c=5,$ and $a\ge 1.$ Let $d=a-1.$ Then $d,$ $b,$ and $c$ are all nonnegative integers with $d+b+c=4.$ We can view this as placing two dividers among four dots, which can be done in a total of $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=\boxed{15}$ ways.