<s>[INST] Square coloring problem only using 2 colors 
"We need to color $4×4$ square using $4$ black color and $12$ white color. Then, how many cases it may be? Flip is prohibited but rotating is ok"

I tried case by case (inner square and rest) anf obtained answer 389. But I don't know if it's correct. Help me please.
 [/INST] We apply PET (Polya Enumeration Theorem) here and this needs the cycle
index.  There are  four rotations.  The  first is  the identity  which
contributes
$$a_1^{16}.$$
There are  the rotations by $90$  degrees and by $270$  degrees, which
contribute
$$2 a_4^4.$$
The rotation by $180$ degrees contributes
$$a_2^8.$$
This yields the cycle index
$$Z(G) = \frac{1}{4} (a_1^{16} + 2 a_4^4 + a_2^8).$$
We thus have for the desired quantity
$$[B^4 W^{12}] Z(G; B + W)
\\ = [B^4 W^{12}]
\frac{1}{4} ((B+W)^{16} + 2 (B^4+W^4)^4 + (B^2+W^2)^8)
\\ = \frac{1}{4} {16\choose 4} +
\frac{1}{2} [B W^3] (B+W)^4 + \frac{1}{4} [B^2 W^6] (B+W)^8
\\ = \frac{1}{4} {16\choose 4} + \frac{1}{2} {4\choose 1}
+ \frac{1}{4} {8\choose 2}.$$
This yields
$$\bbox[5px,border:2px solid #00A000]{464}$$
confirming the data from the comment.
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