You can put this solution on YOUR website! To express a permutation as a product of transpositions, we need to identify the pairs of elements that need to be swapped to transform the identity permutation into the given permutation.
**1. [1 2 3 4 3 4 5 6 5 6 8 2 7 8 7 1 ]**
We can break down this permutation into the following steps:
1. Swap 2 and 8: (2 8)
2. Swap 1 and 7: (1 7)
3. Swap 3 and 4: (3 4)
4. Swap 5 and 6: (5 6)
So, the permutation can be expressed as a product of transpositions:
```
(2 8)(1 7)(3 4)(5 6)
```
**2. [1 2 1 3 3 4 8 4 5 6 6 7 7 8 5 2 ]**
We can break down this permutation into the following steps:
1. Swap 2 and 3: (2 3)
2. Swap 4 and 8: (4 8)
3. Swap 6 and 7: (6 7)
4. Swap 5 and 2: (5 2)
So, the permutation can be expressed as a product of transpositions:
```
(5 2)(6 7)(4 8)(2 3)
```
