Joanna has seven beads that she wants to assemble into a bracelet. Five of the beads have the same color, and the other two all have different colors. In how many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical if one can be rotated and/or reflected to obtain the other.)

blackpanther Feb 14, 2024

#2**0 **

Lets start by assembling the beads in a staight line:

There are \(\frac{7!}{5!}=42\) number of ways to arrange them in a straight line (\(7!\) for the number os beads and divide by \(5!\) for the number of same beads).

Now to divide due to rotation and reflection:

To do this we divide by 7 for the number of rotations and by 2 for the mirror image:

therefore:

\(42/14=\boxed{3}\)

ggwp

EnormousBighead Feb 15, 2024