$15 \cdot 2$ In the set $\{1000,1001, \cdots, 2000\}$, how many pairs of consecutive integers can be added without carrying?

(10th American Invitational Mathematics Examination, 1992)
[Solution] Let $n$ be represented as $\overline{1 a b c}$.
If one of $a, b, c$ is $5, 6, 7$ or 8, then adding $n$ and $n+1$ requires a carry; if $b=9, c \neq 9$, or $a=9, b$ and $c$ are at least one not 9, then adding $n$ and $n+1$ also requires a carry.
If $n$ is not as described above, then it must be one of the following forms:
$$
\overline{1 a b c}, \overline{1 a b 9}, \overline{1 a 99}, 1999 \text{. }
$$

where $a, b, c \in\{0,1,2,3,4\}$.
For such $n$, adding $n$ and $n+1$ does not require a carry, so there are
$$
5^{3}+5^{2}+5+1=156 \text{ pairs. }
$$