Question

A bracelet is made from exactly 5 beads each of a different color. Two bracelets created from this collection of beads are considered the same if for each bead the colors of the beads to its left and right are the same. How many different bracelets can be made?

Answer 1

24

Explanation:

Let's consider the possible combinations for the beads on the bracelet while adhering to the given conditions.

Since each bracelet consists of exactly 5 beads, we can place the first bead in any color, as it has no left or right neighbors. For the subsequent beads, each bead's color must be different from both its left and right neighbors to satisfy the given condition.

After placing the first bead, we have 4 options for the second bead (any color except the one chosen for the first bead). Then, for the third bead, we have 3 options (any color except the two colors chosen for the first two beads).

For the fourth bead, we have 2 options left (any color except the three colors chosen for the previous beads). Finally, for the fifth and last bead, there is only 1 option left.

So, the total number of different bracelets that can be made is the product of the number of choices for each bead:

Total number of bracelets = \(4 \times 3 \times 2 \times 1 = 24.\)

Therefore, 24 different bracelets can be made using 5 beads of different colors, where two bracelets are considered the same if the colors of the beads to their left and right are the same.