jmc

Note that a $7$ digit increasing integer is determined once we select a set of $7$ digits. To determine the number of sets of $7$ digits, consider $9$ urns labeled $1,2,\cdots ,9$ (note that $0$ is not a permissible digit); then we wish to drop $7$ balls into these urns. Using the ball-and-urn argument, having $9$ urns is equivalent to $8$ dividers, and there are $\left(\genfrac{}{}{0ex}{}{8+7}{7}\right)=\left(\genfrac{}{}{0ex}{}{15}{7}\right)=6435\equiv \boxed{435}(\mathrm{mod}1000)$.