Problem 788

Dominating Numbers

A dominating number is a positive integer that has more than half of its digits equal.

For example, $2022$ is a dominating number because three of its four digits are equal to $2$ . But $2021$ is not a dominating number.

Let $D (N)$ be how many dominating numbers are less than $10^{N}$ . For example, $D (4) = 603$ and $D (10) = 21893256$ .

Find $D (2022)$ . Give your answer modulo $1 000 000 007$ .

支配数是指有一半以上数字相同的正整数。

例如， $2022$ 是支配数，因为其四个数字中有三个是 $2$ 。反之， $2021$ 不是一个支配数。

记 $D (N)$ 为小于 $10^{N}$ 的支配数的数目。例如， $D (4) = 603$ ， $D (10) = 21893256$ 。

求 $D (2022)$ ，并将你的答案对 $1 000 000 007$ 取余。