Lucky numbers

First, I would like to find the number of lucky number if $\overline{ABC} = \overline{DEF}$ .

Since $\overline{ABC} = \overline{DEF}$ , we simply find the possible number of $\overline{ABC}$ which is $9 \times 10 \times 10 = 900$ .

The possible number of $G$ is 10, so the number of lucky number if $\overline{ABC} = \overline{DEF}$ is $900 \times 10 = 9000$ .

For $\overline{ABC} = \overline{EFG}$ , we find the possible number of $\overline{ABC}$ which is $900$ .

The possible number of $D$ is 10, so the number of lucky number if $\overline{ABC} = \overline{DEF}$ is $900 \times 10 = 9000$ .

There must be the same lucky number we get for $\overline{ABC} = \overline{DEF}$ and $\overline{ABC} = \overline{EFG}$ which are $1111111, 2222222, 3333333, \ldots 9999999$ .

So the number of all lucky numbers is $9000+9000-9 = \boxed{17991}$