## Task A-2.5.

How many different bracelets can be made consisting of four black and four white beads arranged in a circle? Two bracelets are considered different if they cannot be rotated to have the same sequence of beads.
## Solution.

Let's consider how long the longest string consisting only of black balls is.

If it is 4 in length, then the remaining four balls are obviously white, and this is the only possibility in this case.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-10.jpg?height=180&width=240&top_left_y=715&top_left_x=834)

If it is 3 in length, then these three black balls are definitely surrounded by two whites, otherwise the string would be 4 in length. We need to place one more black ball in one of the three remaining positions. We have two different possibilities, as shown in the figure:
![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-10.jpg?height=182&width=514&top_left_y=1145&top_left_x=694)

If it is 2 in length, then the remaining two black balls may or may not be adjacent.

If they are adjacent, we have two different possibilities, as shown in the figure:
![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-10.jpg?height=180&width=506&top_left_y=1545&top_left_x=702)

If they are not adjacent, we have two possibilities, as shown in the figure:
![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-10.jpg?height=186&width=512&top_left_y=1854&top_left_x=697)

If the longest described string is 1 in length, then we have only one bracelet, and that is the one where black and white balls alternate.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-10.jpg?height=177&width=237&top_left_y=2242&top_left_x=835)

Therefore, there are a total of eight different bracelets.