Your making a necklace with 4 red peices and 4 yellow peices. How many different combination of necklaces can you make?
is 2^8 correct.
Hello, harold3331!
This is a very tricky problem . . .
You're making a necklace with 4 red pieces and 4 yellow pieces.
How many different combination of necklaces can you make?
is 2^8 correct? . . . . no
Your answer assumes there are at least eight of each color
. . and includes all the combinations from $\displaystyle RRRRRRRR\text{ to }YYYYYYYY.$
Even worse, this is a necklace . . . a circular arrangement.
So that, for example: .$\displaystyle RRRRYYYY \:=\:RRYYYYRR$
. . That is: . $\displaystyle \begin{array}{cccc}& R\;R \\ Y & & R \\ Y & & R \\ & Y\;Y \end{array}\quad=$ . .$\displaystyle \begin{array}{ccc}& R\;R \\ R & & Y \\ R & & Y \\ & Y\;Y \end{array}$
There is one more "trap" in a Necklace Problem, overlooked by the best of us.
This is similar to seating 4 men and 4 women around a circilar table,
. . except a necklace can be "flipped over".
So that: . $\displaystyle \begin{array}{ccc} & R:R \\ Y &: & R \\ Y &:& Y \\ & Y:R\end{array}$ . .$\displaystyle = \quad \begin{array}{ccc}& R:R \\ R &:& Y \\ Y &:& Y \\ & R:Y\end{array}$
Care to try again?