Introduction to Circular Permutations
When arranging objects in a line (linear permutation), every position is distinct. However, when arranging objects in a circle, we encounter a different scenario. In a circle, there is no 'start' or 'end'—only the relative positions of the items matter. Rotating the circle does not create a new arrangement.
The Formula
For $n$ distinct objects arranged in a linear fashion, the number of permutations is $n!$.
In a circular arrangement, we fix one object to eliminate rotational symmetry. Once one object is fixed, the remaining $n-1$ objects are effectively arranged in a line relative to the fixed object. Therefore, the number of ways to arrange $n$ distinct items in a circle is:
$$(n - 1)!$$
Solving the Problem
We need to seat 7 students in a circle.
- Identify the values: Here, $n = 7$.
- Apply the formula: The number of ways is $(7 - 1)!$.
- Calculate: $(6)! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $6! = 720$
There are 720 ways to seat 7 students in a circle.
Why $(n-1)!$?
Imagine you have 7 chairs. If you rotate everyone one seat to the left, the relative neighbors remain the same. To account for this, we 'fix' one student in a seat, which breaks the symmetry, turning the problem into arranging the remaining 6 students in the 6 remaining chairs, which is $6!$.