Ytukyg

If $n$ is the number of necklaces which can be formed using $17$ identical pearls and two identical diamonds and similarly $m$ is the number of necklaces which can be formed using $17$ identical pearls and $2$ different diamonds, then the value of $m$ and $n$ is?

Use Burnside's Lemma.

If $G$ is a finite group of permutations on a set $S$, then the number of orbits of $G$ on $S$ is
begin{align}
frac{1}{|G|}sum_{g in G}|operatorname{fix}(g)|.
end{align}

Observe there are $binom{19}{2}=171$ possible designs (of course, some of them are equivalent by rotation and reflection). Let $G = D_{38}$ act on the $171$ necklace designs. Observe we see that
begin{matrix}
text{element} & text{Number of Designs Fixed by Element}\
e & 171\
r^k, k=1,ldots, 18 & 0\
f & 9\
fr^k, k=1,ldots, 18 & 9
end{matrix}
Hence by Burnside's lemma, we see that
begin{align}
frac{1}{38}(171+9+9*18) = 9.
end{align}

Indeed, one possible configuration is that the two diamonds are $1$ apart on one side and $16$ apart on the other side or $3$ apart on one side and $14$ apart on the other side...until $17$ apart on one side and $0$ apart on the other side. Hence $9$ configurations.