Math Problem Statement
et be a prime. Suppose that the number of ways to place indistinguishable red marbles, indistinguishable green marbles, and indistinguishable blue marbles around a circle such that no red marble is next to a green marble and no blue marble is next to a blue marble is . (Rotations and reflections of the same configuration are considered distinct.) Given that , where is a non-negative integer and is not divisible by , and is the remainder of when divided by , compute .
Solution
We are given a problem involving the placement of indistinguishable red marbles, indistinguishable green marbles, and indistinguishable blue marbles around a circle with the following restrictions:
- No red marble is next to a green marble.
- No blue marble is next to another blue marble.
Step 1: Analyze the problem setup
We are dealing with a circular arrangement, so rotations and reflections of configurations are considered the same. The total number of marbles is , with each of the three colors (red, green, and blue) having marbles. We need to count how many distinct arrangements of these marbles can be made while satisfying the given conditions.
Step 2: Restrictions on adjacent marbles
- The red marbles cannot be adjacent to green marbles.
- The blue marbles cannot be adjacent to other blue marbles.
Thus, adjacent positions in the circle must alternate between the different colors, respecting these restrictions. Since no two red marbles can be adjacent to green marbles, we will place the red marbles first and then place the other two colors (green and blue) in the remaining spaces.
Step 3: Using the "stars and bars" method
A key observation here is that the red marbles create gaps between them where green and blue marbles can be placed. For red marbles arranged around a circle, there are exactly gaps (since the arrangement is circular).
The remaining problem reduces to placing green and blue marbles in these gaps, subject to the condition that no blue marble can be adjacent to another blue marble. This is equivalent to distributing the green and blue marbles in such a way that no two blue marbles are adjacent.
Step 4: Counting the number of valid arrangements
By the restrictions on adjacency, we can think of the problem as arranging the blue marbles in the gaps between the green marbles. Since we already have green marbles and there are available gaps, the number of ways to place the blue marbles is simply the number of ways to choose gaps from the available positions.
This is a straightforward combinatorial problem, and the number of valid arrangements is , which is the total number of ways to place the marbles around the circle under the given restrictions.
Step 5: Expressing
Given the number of ways to arrange the marbles is , this expression matches the form where and (since there are no additional factors). Thus, we have:
- ,
- .
Step 6: Compute
The remainder of when divided by is because and .
Now, we compute:
Thus, the final answer is:
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Circular Arrangements
Stars and Bars
Modular Arithmetic
Formulas
N = p^p
N = p^m · n
pm + r = p^2 + 1
Theorems
Circular Arrangement Theorem
Stars and Bars Method
Modular Arithmetic
Suitable Grade Level
College-level
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