1. How many pairs $(n, r)$ are there in the array satisfying $0 \leqslant r \leqslant n \leqslant 63$ for which the binomial coefficient $\mathrm{C}_{n}^{r}$ is even (assuming $\left.\mathrm{C}_{0}^{0}=1\right) ?$
From Example 5, we know that the number of odd numbers in $\mathrm{C}_{n}^{0}, \mathrm{C}_{n}^{1}, \cdots, \mathrm{C}_{n}^{n}$ is $2^{S(n)},$ where $S(n)$ is the sum of the binary digits of $n$.

Since $63=2^{6}-1=(111111)_{2}$, when $0 \leqslant n \leqslant 63$, we have $0 \leqslant S(n) \leqslant 6$.

Classifying and counting by the sum of the binary digits $S(n)$ of $n$, we know that the number of $n$ satisfying $S(n)=k \in\{0,1, \cdots, 6\}$ is $\mathrm{C}_{6}^{k}$. 
Thus, the number of odd binomial coefficients $\mathrm{C}_{n}^{r}$ is
$$
\sum_{k=0}^{6} 2^{k} \mathrm{C}_{6}^{k}=(1+2)^{6}=729 \text {. }
$$

The total number of binomial coefficients is
$$
1+2+\cdots+64=2080 \text {. }
$$

Therefore, the number of the required arrays is
$$
2080-729=1351 .
$$