<s>[INST] Burnsides Theorem Application - Necklace Having a lot of trouble understanding this question: How many different necklaces can be made from eight black beads and four white beads?
What I know:
Burnsides Theorem is given by: $$\frac{1}{|G|} \sum_{g \in G} |\operatorname{fix}_{\Omega} (g)|$$
8+4=12 and this corresponds to the Dihedral group of order 12 (I use the notation $D_{2n})$ so I will note this as $D_{24}$, also known as the symmetric 12-gon. 
So we know that $|G|=24$ and the total number of necklace configurations is given by $\binom{12}{4}=495$ so I'm guessing we have to $Fix(e)=495$? But I'm unsure how to apply the rest of Burnsides theorem from here and work out the configurations of rotations and reflections. 
Any help would be great, thanks.
 [/INST] Given that  we have dihedral symmetry  we suppose we are  working with
bracelets (naming convention by OEIS).   This requires the cycle index
$Z(D_{12})$ of  the dihedral  group $D_{12}.$ We  have for  the cyclic
group that
$$Z(C_{12}) = \sum_{d|12} \varphi(d) a_d^{12/d}
= \frac{1}{12}
(a_1^{12} + a_2^6 + 2 a_3^4 + 2 a_4^3 + 2 a_6^2 + 4  a_{12}).$$
We get twelve  more permutations corresponding to flips  about an axis
passing through opposite slots or opposite edges, for a result of
$$Z(D_{12}) = \frac{1}{24}
(a_1^{12} + a_2^6 + 2 a_3^4 + 2 a_4^3 + 2 a_6^2 + 4  a_{12})
+ \frac{1}{24} ( 6 a_1^2 a_2^5 + 6 a_2^6).$$
By  the Polya  Enumeration  Theorem  (PET) we  are  interested in  the
quantity
$$[B^8 W^4] Z(D_{12}; B+W).$$
Working through the terms we find

 $[B^8 W^4] (B+W)^{12} = {12\choose 8},$ 
 $[B^8 W^4] (B^2+W^2)^{6} = [B^4 W^2] (B+W)^{6}
=  {6\choose 4},$ 
 $[B^8 W^4] 2 (B^3+W^3)^{4} = 0$ 
 $[B^8 W^4] 2 (B^4+W^4)^{3} = [B^2 W] 2 (B+W)^{3}
= 2 {3\choose 2}$ 
 $[B^8 W^4] 2 (B^6+W^6)^{2} = 0$ 
 $[B^8 W^4] 4 (B^{12}+W^{12}) = 0$ 

We get from the reflections

 $[B^8 W^4] 6 (B+W)^2 (B^2+W^2)^5
\\ = [B^6 W^4] 6 (B^2+W^2)^5 + [B^7 W^3] 12 (B^2+W^2)^5 +
[B^8 W^2] 6 (B^2+W^2)^5
\\ = [B^3 W^2] 6 (B+W)^5 + [B^4 W] 6 (B+W)^5
\\ = 6 {5\choose 3} + 6 {5\choose 4}$ 
 $[B^8 W^4] 6 (B^2+W^2)^{6} = [B^4 W^2] 6 (B+W)^{6}
=  6 {6\choose 4}.$ 

Collecting everything we find for our result that it is
$$\frac{1}{24} \left({12\choose 8} + {6\choose 4} + 2 {3\choose 2}
+ 6{5\choose 3} + 6{5\choose 4} + 6 {6\choose 4}\right).$$
which yields
$$\bbox[5px,border:2px solid #00A000]{29.}$$
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