Begins with 1, ends in 2: .\(\displaystyle 1\;\_\;\_\;\_\;\_\;\_\;\_\;2\)
. . The middle digits \(\displaystyle \{3,3,3,6,6,8\}\) can be arranged in \(\displaystyle \frac{6!}{3!2!} = 60\) ways.
Begins with 1, ends in 6: ..\(\displaystyle 1\;\_\;\_\;\_\;\_\;\_\;\_\;6\)
. . The middle digits \(\displaystyle \{2,3,3,3,6,8\}\) can be arranged in \(\displaystyle \frac{6!}{3!} = 120\) ways.
Begins with 1, ends in 8: ..\(\displaystyle 1\;\_\;\_\;\_\;\_\;\_\;\_\;8\)
. . The middle digits \(\displaystyle \{2,3,3,3,6,6\}\) can be arranged in \(\displaystyle \frac{6!}{3!2!} = 60\) ways.
Begins with 2, ends in 6: ..\(\displaystyle 2\;\_\;\_\;\_\;\_\;\_\;\_\;6\)
. . The middle digits \(\displaystyle \{1,3,3,3,6,8\}\) can be arranged in \(\displaystyle \frac{6!}{3!} = 120\) ways.
Begins with 2, ends in 8: ..\(\displaystyle 2\;\_\;\_\;\_\;\_\;\_\;\_\;8\)
. . The middle digits \(\displaystyle \{1,3,3,3,6,6\}\) can be arranged in \(\displaystyle \frac{6!}{3!2!} = 60\) ways.
Begins with 3, ends in 2: ..\(\displaystyle 3\;\_\;\_\;\_\;\_\;\_\;\_\;2\)
. . The middle digits \(\displaystyle \{1,3,3,6,6,8\}\) can be arranged in \(\displaystyle \frac{6!}{2!2!} = 180\) ways.
Begins with 3, ends in 6: ..\(\displaystyle 3\;\_\;\_\;\_\;\_\;\_\;\_\;6\)
. . The middle digits \(\displaystyle \{1,2,3,3,6,8\}\) can be arranged in \(\displaystyle \frac{6!}{2!} = 360\) ways.
Begins with 3, ends in 8: ..\(\displaystyle 3\;\_\;\_\;\_\;\_\;\_\;\_\;8\)
. . The middle digits \(\displaystyle \{1,2,3,3,6,6\}\) can be arranged in \(\displaystyle \frac{6!}{2!2!} = 180\) ways.
Therefore, there are:
. . \(\displaystyle 60+120+60+120+60+180+360+180 \:=\:1140\) such numbers.