This is not a home work question, I'm preparing for an entrance test.
The number of different necklaces you can form with $2$ black and $6$ white beads is?
My approach: We can place the white beads in the necklace in $1$ way because they all are white. Then the black beads can again be placed anywhere in $1$ way, once a black bead is placed it now acts like a reference, and I can place the last black bead either next to the first black, or with a gap of $1,2,3$ white beads, giving me $4$ combinations as, wbbwwwww, wbwbwwww, wbwwbwww, wbwwwbww.
Is there any other better approach for this?
I tried to follow this way, How many different necklaces can be formed with $6$ white and $5$ red beads? but I am getting fractional values,
$$\frac{7!}{6! \cdot 2! \cdot 2}$$
Why is it happening this way? Can't we use this formula logic in all cases?