Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will not happen is given by  m/n where m and n are relatively prime positive integers. Find m + n. (right marks 10.00)

The problem involves six people sitting in a row of six chairs. Each person can either place their feet on the floor, cross their legs to the right, or cross their legs to the left. We are tasked with finding the probability that no two adjacent people will cross their legs towards each other, which would create a conflict.

### Step 1: Define the leg positions

Each person can be in one of three states:
1. Feet on the floor (denoted as "F"),
2. Legs crossed to the right (denoted as "R"),
3. Legs crossed to the left (denoted as "L").

The total number of configurations without any restrictions is \( 3^6 \), since each of the 6 people has 3 possible choices.

### Step 2: Identify the "conflict" situation

A conflict occurs when a person crosses their legs to the left (L) and the person directly to their right crosses their legs to the right (R). Specifically, this conflict occurs if two adjacent people are seated as:
\[
L R
\]
We need to calculate how many valid seating arrangements exist where this configuration does not appear anywhere among adjacent pairs.

### Step 3: Use of recurrence relation

To count the valid seating arrangements, define \( a_n \) as the number of valid configurations for \( n \) people. We can develop a recurrence relation based on the following observations:

- If the first person has their feet on the floor ("F"), then the remaining \( n-1 \) people can be arranged in any valid configuration, so there are \( a_{n-1} \) ways to complete the seating.
- If the first person crosses their legs to the right ("R"), the next person must not cross their legs to the left (as that would create a conflict), so the second person must either place their feet on the floor ("F") or cross their legs to the right ("R"). This leaves \( a_{n-1} \) valid ways to complete the seating.
- If the first person crosses their legs to the left ("L"), the next person must not cross their legs to the right, so the second person must either place their feet on the floor ("F") or cross their legs to the left ("L"). This again leaves \( a_{n-1} \) valid ways to complete the seating.

Thus, the recurrence relation is:
\[
a_n = 2a_{n-1} + 2a_{n-2}
\]
where:
- \( 2a_{n-1} \) accounts for the cases where the first person places their feet on the floor or crosses their legs to the right or left, and
- \( a_{n-2} \) accounts for cases where the first two persons don't conflict.



Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will not happen is given by m/n where m and n are relatively prime positive integers. Find m + n.

Explore a probability problem involving six people randomly seating themselves in six chairs, with three choices for leg positions: feet on the floor, legs crossed to the right, or to the left. Learn how to solve using recurrence relations and find the probability that no conflict occurs between adjacent people. Suitable for Grades 11-12 and college-level students.

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