Regular Expression C# Random Letter from specific List of Letters

I want the string (length 8) format to be one letter from a specific letter list A, B, C (neither of these letters should be repeated) the letters have a random position and the rest are numbers (0-9). For example:

1234A567  - Valid
277897C0  - Valid
A100299B  - Not valid
12C3879C  - Not valid

I tried something like this: (\d){7}(A|C|E|F|G|H|J|K|M|U|Z){1} but this does not work. Help?