34. Find the number of 20 -tuples of integers $x_{1}, \ldots, x_{10}, y_{1}, \ldots, y_{10}$ with the following properties:
- $1 \leq x_{i} \leq 10$ and $1 \leq y_{i} \leq 10$ for each $i$;
- $x_{i} \leq x_{i+1}$ for $i=1, \ldots, 9$
- if $x_{i}=x_{i+1}$, then $y_{i} \leq y_{i+1}$.
Solution: $\left.\begin{array}{c}109 \\ 10\end{array}\right)$
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.