$S$ = {necklaces of length 14 with 8 blue, 3 green and 3 brown beads}.
Clearly,$|S| = \frac {14!} {8!3!3!} $.
The group of symmetries, $G = D_{14}$. Clearly, $|G| = 28$. And as you have identified G is acting on S.
And by Burnside lemma, required answer is, $$ \#orbits = \frac 1 {|G|}*\sum_{\sigma \in G}fix(\sigma)$$
1) $\sigma = identity $
Then it fixes any element in $S$. Thus, $fix(\sigma) = |S| = \frac {14!} {8!3!3!}$
2) Rotations, $\sigma$ = rotation by $\frac {360} {14} degree$ clockwise = $p^1$
Carefully consider the cyclic structure of the permutation $\sigma$,
$$\begin{pmatrix}
1&2&3&4&5&6&7&8&9&10&11&12&13&14\\
2&3&4&5&6&7&8&9&10&11&12&13&14&1
\end{pmatrix}$$
$$\equiv (1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14) \ \ \text{[in cycle notation]}$$
If $\sigma$ fixes $x \in S$, then all vertex of the 14-gon must have the same color which is not the case, hence $fix(\sigma)=0$.
Clearly, $fix(\sigma=p^{13})=0$.
3)$\sigma=p^2$ i.e
$$\begin{pmatrix}
1&2&3&4&5&6&7&8&9&10&11&12&13&14\\
3&4&5&6&7&8&9&10&11&12&13&14&1&2
\end{pmatrix}$$
$$\equiv (1\ 3\ 5\ 7\ 9\ 11\ 13)(2\ 4\ 6\ 8\ 10\ 12\ 14) \ \ \text{[in cycle notation]}$$.
So, $\sigma$ has two cycles of length 7 and if you think carefully, for x to be in $fix(\sigma)$ all vertices in a single cycle will have same bead color. Which is not possible with 8 red, 3 blue and 3 brown beads. So, $fix(\sigma)=0$.
Clearly, $fix(\sigma=p^{12})=0$.
In the same fashion compute the $fix(\sigma)$ for the remaining by looking into the cycle structure and then use the burnside to get the required answer.
To count the favorable cases, observe that if you select a red shirt and a blue scarf among the four selected shirts, two selected pants, and three selected scarves, you must select three of the other five available shirts, two of the four available pants, and two of the other six available scarves. Therefore, the number of favorable cases is
$$\binom{5}{3}\binom{4}{2}\binom{6}{2}$$
Hence, the probability of selecting a red shirt and a blue scarf when four shirts, two pants, and three scarves are chosen from six different color shirts, four different color pants, and seven different color scarves is
$$\frac{\dbinom{5}{3}\dbinom{4}{2}\dbinom{6}{2}}{\dbinom{6}{4}\dbinom{4}{2}\dbinom{7}{3}}$$
Best Answer
Since it doesn't matter where you place the first bead, the number of ways of arranging the other beads = $6!$
But considering the fact that the beads doesn't have a differentiation between left/right, the final answer would be: $$\dfrac{6!}{2}$$