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.

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.
