To solve the problem of finding the number of distinct colorings of a cube using 6 distinct colors (each face a unique color), we use Burnside's Lemma, which calculates the average number of fixed colorings under all symmetries of the cube.

Key Observations:

• The symmetry group of the cube has 24 elements (6 choices for front face × 4 choices for right face).
• For distinct colorings, most symmetries have no fixed colorings (since they require repeated colors). Only the identity symmetry fixes all colorings.

Calculations with Burnside's Lemma:

• Identity symmetry: Fixes all (6! = 720) colorings.
• All other symmetries: (90°/270° face rotations, 180° face rotations, vertex rotations, edge rotations) → 0 fixed colorings (since distinct colors can't repeat).

Average fixed colorings = (\frac{720}{24} = 30).

Answer: (\boxed{30})

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