Question 1205220
To determine the order of a permutation, we need to find the smallest positive integer k such that the permutation raised to the power of k equals the identity permutation.

**a) [1 2 3 4 5 2 3 4 5 1]**

This permutation can be written in cycle notation as (1 2 3 4 5). 
* Applying this permutation 5 times, we get back to the original order. 
* Therefore, the order of this permutation is 5.

**b) [1 2 3 4 4 3 1 2 ]**

This permutation can be written in cycle notation as (1 2)(3 4). 
* Applying (1 2) twice, we get back to the original order.
* Similarly, applying (3 4) twice, we get back to the original order.
* The least common multiple of 2 and 2 is 2.
* Therefore, the order of this permutation is 2.