Math Problem Statement

et p=2017p = 2017 be a prime. Suppose that the number of ways to place pp indistinguishable red marbles, pp indistinguishable green marbles, and pp 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 NN. (Rotations and reflections of the same configuration are considered distinct.) Given that N=pmnN = p^m \cdot n, where mm is a non-negative integer and nn is not divisible by pp, and rr is the remainder of nn when divided by pp, compute pm + rpm + r.

Solution

We are given a problem involving the placement of pp indistinguishable red marbles, pp indistinguishable green marbles, and pp 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 3p3p, with each of the three colors (red, green, and blue) having pp 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 pp red marbles arranged around a circle, there are exactly pp gaps (since the arrangement is circular).

The remaining problem reduces to placing pp green and pp blue marbles in these pp 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 pp blue marbles in the gaps between the green marbles. Since we already have pp green marbles and there are pp available gaps, the number of ways to place the blue marbles is simply the number of ways to choose pp gaps from the pp available positions.

This is a straightforward combinatorial problem, and the number of valid arrangements is ppp^p, which is the total number of ways to place the marbles around the circle under the given restrictions.

Step 5: Expressing NN

Given the number of ways to arrange the marbles is N=ppN = p^p, this expression matches the form N=pmnN = p^m \cdot n where m=pm = p and n=1n = 1 (since there are no additional factors). Thus, we have:

  • m=pm = p,
  • n=1n = 1.

Step 6: Compute pm+rpm + r

The remainder of nn when divided by pp is r=1r = 1 because n=1n = 1 and 1modp=11 \mod p = 1.

Now, we compute:

pm+r=pp+1=p2+1.pm + r = p \cdot p + 1 = p^2 + 1.

Thus, the final answer is:

p2+1.\boxed{p^2 + 1}.

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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