Binomial[12 - u - v - w + z,  4 - u - w + z] /.
{u -> Subscript[c, [Alpha][Beta]],
v -> Subscript[c, [Beta][Gamma]], w -> Subscript[c, [Gamma][Alpha]],
z -> Subscript[c, [Alpha][Beta][Gamma]]}

Binomial[12 - Subscript[c, [Alpha][Beta]]
+ Subscript[ c, [Alpha][Beta][Gamma]] - Subscript[c, [Beta][Gamma]]
- Subscript[c, [Gamma][Alpha]], 4 - Subscript[c, [Alpha][Beta]]
+ Subscript[ c, [Alpha][Beta][Gamma]] - Subscript[c, [Gamma][Alpha]]]