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The following represents the group of rigid motions of some figure with 6 vertices. G = {(1)(2)(3)(4)(5)(6), (12)(3456), (1)(2)(35)(46), (12)(3654)} The pattern inventory for colors

The following represents the group of rigid motions of some figure with 6 vertices. G = {(1)(2)(3)(4)(5)(6), (12)(3456), (1)(2)(35)(46), (12)(3654)}

The pattern inventory for colors red and white is r6 + 2r5w + 5r4w2 + 6r3w3 + 5r2w4 + 2rw5 + w6. How many distinguishable colorings have at least 3 red vertices?

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