Find the number of different colorings of the faces of the cube with 2 white, 1 black, 3 red faces? I tried using Polya theorem same in this https://nosarthur.github.io/math%20and%20physics/2017/08/10/polya-enumeration.html guide, using S4 group for cube faces. But i Have a $\frac{1}{24}12w^2*b*r^3$.
Please help to me how count coloring faces of the cube and other platonic solids.

First consider the three red faces.  Either one of them is opposite another, forming three-quarters of a loop, or all three are adjacent to the same vertex.  In the former case, there are three possibilities for the black face; it either "completes the loop," it's to the right of the loop's open face, or it's to the left of the loop's open face.  The two remaining faces must be white.
Edited to add:  However, as noted in the comments, the latter two possibilities are actually the same coloring, rotated a half turn around the axis through the center of the loop's open face.  Therefore, there are only two distinct colorings in this family.
If all three red faces are adjacent to the same vertex, then the two white faces will complete a loop with two of the red faces.  Use the existing coloring to choose an orientation.  The black face can be on either side of the resulting loop, resulting in two possibilities.  Edited to add:  But these are the same.  Assume the two white faces are "near" and "top" and the black face is "right."  Rotate the die a quarter-rotation toward yourself around the left-right axis (so that the two white faces are now "bottom" and "near") and then flip the die a half-rotation around the near-far axis (bringing the "bottom" face to "top" and the "right" face to the "left").
There are, therefore, five (edit: three) different colorings.
