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The problem asks us to determine whether the permutation 
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
3 & 1 & 4 & 5 & 2 
\end{pmatrix}
\]
is even or odd.

### Step 1: Understanding the Permutation

This permutation can be written as:
- \( 1 \to 3 \)
- \( 2 \to 1 \)
- \( 3 \to 4 \)
- \( 4 \to 5 \)
- \( 5 \to 2 \)

### Step 2: Writing the Permutation in Cycles

We can decompose the permutation into disjoint cycles:
- Start with \( 1 \). Since \( 1 \to 3 \), and \( 3 \to 4 \), and \( 4 \to 5 \), and \( 5 \to 2 \), and \( 2 \to 1 \), we have the cycle:
  \[
  (1 \ 3 \ 4 \ 5 \ 2)
  \]

Thus, the permutation consists of a single cycle:
\[
(1 \ 3 \ 4 \ 5 \ 2)
\]

### Step 3: Determining Parity

To determine whether a permutation is even or odd, we need to check the length of the cycles. A cycle of length \( k \) corresponds to \( k-1 \) transpositions.

The cycle \( (1 \ 3 \ 4 \ 5 \ 2) \) has a length of 5, so it corresponds to \( 5 - 1 = 4 \) transpositions.

Since the number of transpositions is even, the permutation is **even**.

### Final Answer: The permutation is even.

Do you have any questions, or would you like further details?

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Here are 5 related questions:
1. What is the difference between even and odd permutations?
2. How do transpositions relate to the parity of a permutation?
3. Can a permutation with multiple disjoint cycles be both even and odd?
4. How do you determine the number of transpositions for a cycle?
5. What are some real-world applications of permutation theory?

**Tip:** When decomposing permutations into cycles, always make sure the cycles are disjoint. This helps simplify the calculation of parity.

Find whether the permutation (1 2 3 4 5)(3 1 4 5 2) is even or odd.

Learn how to determine whether the permutation (1 2 3 4 5)(3 1 4 5 2) is even or odd. This problem covers key concepts in permutations, cycle notation, and the parity theorem. Suitable for students in Grades 9-12.

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