I came across this problem in a book called limits, sequences combinations great book for intro to combinatorics .
A rattle consists of a ring with $3$ white beads and $7$ red ones strung on it. Some rattles seemingly different can be made identical by arranging the rings and moving the beads in a suitable manner (rotation or flipping). Find the number of different rattles .
I of course thought polya enumeration on this one but was thinking it can be done case by case without being too messy . Can anyone help? Also this is essentially the same problem as a necklace with $n$ beads, $k$ colors is it not ?