Joanna has six beads that she wants to assemble into a bracelet. Two of the beads have the same color, and the other four all have the same color. 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.)
Let's consider the two cases separately:
1 - Two beads have the same color: In this case, we can choose the color for the two identical beads in 4 ways (since there are 4 colors to choose from for those beads), and we can choose the color for the remaining 4 beads in 3 ways (since we cannot choose the same color as the two identical beads). Once we have chosen the colors, there are 5 ways to arrange the beads on the bracelet (since we can rotate it). Therefore, the total number of ways to assemble the bracelet in this case is 4 x 3 x 5 = 60.
2 - Four beads have the same color: In this case, we can choose the color for the four identical beads in 4 ways, and we can choose the colors for the remaining two beads in 3 x 2 = 6 ways (since we cannot choose the same color as the four identical beads). Once we have chosen the colors, there are 3 ways to arrange the beads on the bracelet (since we can rotate it). Therefore, the total number of ways to assemble the bracelet in this case is 4 x 6 x 3 = 72.
Finally, we add the two cases to get the total number of ways: Total number of ways = 60 + 72 = 132
Therefore, there are 132 different ways Joanna can assemble her bracelet.
