Solution:
By setting $z_{i}=10 x_{i}+y_{i}$, we see that the problem is equivalent to choosing a nondecreasing sequence of numbers $z_{1}, z_{2}, \ldots, z_{10}$ from the values $11,12, \ldots, 110$. Making a further substitution by setting $w_{i}=z_{i}-11+i$, we see that the problem is equivalent to choosing a strictly increasing sequence of numbers $w_{1}, \ldots, w_{10}$ from among the values $1,2, \ldots, 109$. There are $\binom{109}{10}$ ways to do this.