DATA 605 - Week 6 Exercise

Section 3.1 Exercise 6

In arranging people around a circular table, we take into account their seats relative to each other, not the actual position of any one person. Show that n people can be arranged around a circular table in (n − 1)! ways.

Intuitively, we know that the first person has only 1 way to sit since all seats and rotations are identical options at this point. The next person now has a contraint and there are n-1 ways left for them to sit. The next person will have n-2 options and so on following the familiar factorial pattern.

Mathematically, we can write the above as the following noting that first seat is essentially asking how many ways there are for a person to choose 1 seat in n \({n\choose 1}\) which is 1.

\[ {n\choose 1}\times (n-1)\times(n-2)\times(n-3)...3\times 2 \times 1 \\ = 1 \times (n-1)\times(n-2)\times(n-3)...3\times 2 \times 1 \\ = 1 \times (n-1)! = (n-1)! \]

DATA 605 - Week 6 Exercise

Mael Illien

3/3/2020

Section 3.1 Exercise 6

In arranging people around a circular table, we take into account their seats relative to each other, not the actual position of any one person. Show that n people can be arranged around a circular table in (n − 1)! ways.