smc

Because we have to use at least one dime and one quarter, this part of the selection is forced. Thus, the number of ways to partition $\text{textdollar}10$ in the desired manner is equal to the number of ways to partition $\text{textdollar}10-10\text{cent}-25\text{cent}=\text{textdollar}9.65$ into quarters and dimes without restriction. However, $\text{textdollar}9.65$ is not a multiple of $25\text{cent}$, so we must use dimes until we reach a multiple of $25\text{cent}$. We first reach such a multiple at $\text{textdollar}9.25$, and, because we have had no freedom up to this point, the number of ways to partition $\text{textdollar}9.25$ into dimes and quarters will yield our desired answer. $\text{textdollar}9.25$ can be partitioned into 37 quarters, and we can replace quarters 2 at a time with 5 dimes to partition it differently. We can replace anywhere from 0 to 18 pairs of quarters with dimes in this manner, giving us $\boxed{19}$ different partitions, which is answer choice $\boxed{19}$.