Compute the number of distinct paths not passing through point (2,2,2) that travel from point (0,0,0) to (4,4,4) in 12 steps, increasing a coordinate by 1 at each step.

Guest was right for the first part, but the number of ways to pass through the point (2, 2, 2) is not \(\frac{6!}{2!2!2!}\), it's \((\frac{6!}{2!2!2!})^2 = 8100\), so the answer is \(\boxed{26550}\)