There are 8 different beads in a box with the colors red, orange, yellow, green, blue, indigo, violet, and white. The red, orange, yellow, green, and blue beads should be together,... There are 8 different beads in a box with the colors red, orange, yellow, green, blue, indigo, violet, and white. The red, orange, yellow, green, and blue beads should be together, but the orange and green beads should not be beside each other. How many ways can you arrange the beads to form a bracelet?
Understand the Problem
The question presents a combinatorial problem involving the arrangement of 8 distinct beads into a bracelet, with specific constraints on the relative positions of certain colored beads. We need to figure out how many ways there are to arrange the beads such that red, orange, yellow, green and blue are together, but orange and green beads are not beside each other. Finally, because the beads are arranged to form a bracelet and not a straight line, we must account for rotational symmetry.
Answer
$216$
Answer for screen readers
$216$
Steps to Solve
- Consider the 5 colored beads as a single unit
Since the red, orange, yellow, green, and blue beads must be together, we can treat them as one unit. This unit, along with the indigo, violet, and white beads, gives us a total of 4 units to arrange.
- Arrange the 4 units
The number of ways to arrange 4 units in a circle is $(4-1)! = 3! = 6$.
- Arrange the 5 colored beads internally
The number of ways to arrange the 5 colored beads linearly is $5! = 120$.
- Account for the orange and green beads not being next to each other
We need to subtract the arrangements where the orange and green beads are next to each other. Treat orange and green as a single unit (OG or GO). Then we have 4 units to arrange: (OG), red, yellow, and blue. The number of ways to arrange these 4 units is $4! = 24$. Since orange and green can be in either order, we multiply by 2, giving $24 \times 2 = 48$ arrangements where orange and green are next to each other.
- Calculate the number of valid arrangements of the 5 colored beads
The number of valid arrangements of the 5 colored beads is the total arrangements minus the invalid arrangements: $120 - 48 = 72$.
- Combine Arrangements
Multiply the number of ways to arrange the 4 units on the bracelet by the number of valid arrangements of the 5 colored beads: $6 \times 72 = 432$.
- Account for bracelet symmetry (flipping)
Since a bracelet can be flipped, we need to divide by 2 to account for the arrangements that are mirror images of each other: $432/2 = 216$.
$216$
More Information
The problem combines circular permutations with constraints on arrangements, requiring careful consideration of the restrictions.
Tips
A common mistake is not accounting for the circular arrangement. Another is forgetting to subtract the cases where orange and green are next to each other. Forgetting to divide by two at the end is also a frequent error.
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