Part 3/5:
Consider the sequence ( 1, 2, 3, \ldots, n ). The first number can occupy ( n ) positions. If we focus on sorting the first number and subsequently shift remaining numbers, the unique ways to sort these can be represented as follows:
- First Arrangement:
Start with ( 1, 2, 3, \ldots, n ).
The first digit can occupy the first position while permutations of subsequent digits occur.
When we rotate the elements, the first number maintains its position, leading to numerous arrangements that depend on the order of the other elements.
- Subsequent Shifts:
- By fixing the first number and shifting the second number, we see how the operations iterate through the remaining digits, leading us to ( (n-1) ) arrangements.