Seating Arrangement Around Various Geometrical Figures

Author: Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal.
We all know the traditional ways to approach a sum where we need to find out different seating arrangements of people around a circular table or in a line.
if we have “N” different things/people and we need to arrange them in a row, the answer will be N!
Simple!
If there are “N” things/people and we need to arrange them in a circular way, the total ways will be (N1)!
Simple Again!
But what if we are asked the seating arrangements of things/people around other geometrical figures ?
Today, in this post we will try to learn how we can arrange different people around these figures.
Note: For all these type of questions we will first try to arrange the first person. Then multiply it with (n1)! Where n is the total number of people that needs to be arranged, since, they can be arranged in that many number of ways.
1. SQUARE :
A square is a geometrical figure that has all its sides equal in length.
We are told that there are 3 chairs on each side of the square, then what are the total number of ways in which we can arrange 12 people around the square table ?
We have this :
We know that the sides of a square are similar and equal (indistinguishable) hence, for the first person, he can sit around the table in 3 ways (any of the 3 chairs put on one side). He can sit either on the left chair, in the middle , or the right chair. Ways to arrange the other people = (N1)! = (121)! = 11!
Hence ways to arrange 12 different persons in the question= 3*(121)! = 3*11!
Shortcut : If there are N people to be arranged around a square table with N/4 seats on one side of it :
Total ways = (N/4) * (N1)!
2. RECTANGLE:
In how many ways can we arrange 10 people around a rectangular table such that there are 3 chairs on the longer side and 2 on the shorter side ?
So what we have from the question is this :
So, if we consider the first person: He can either sit on the shorter side or the longer side.
For Longer Side: there are 3 possibilities: Left, Right and Center
For the Shorter Side: there are 2 possibilities: Left and Right
Hence he can sit in 5 ways
Thus, total arrangements =5* (101)! = 5*9!
Shortcut:
If we have N people that are needed to be arranged around a rectangular table with “a” seats on the longer side and “b” seats on the shorter side the total number of arrangements = (a+b) * (N1)!
3. EQUILATERAL TRIANGLE:
Let there be 12 people who are needed to be arranged around an equilateral triangular table which has 4 seats on each side. How many such arrangements are possible?
From the question we get this figure:
Since for an equilateral triangle all the sides are equal and similar, the first person may sit on any one side.
Ways to sit : 4 (any of the 4 chairs provided)
Hence total number of arrangements of 12 people around a triangular table which is equilateral in nature is = 4*(121)! = 4*11!
Shortcut :
If there are N people that are needed to be arranged around an equilateral triangle, total arrangements = (N/3) *(N1)!
4: ISOSCELES TRIANGLE :
There are 11 people who are needed to be seated around an Isosceles triangle which has 4 seats each on the equal sides and 3 seats on the other. How many Such arrangements are possible ?
From the Question, we get the following figure :
So, for the first person, he can either sit on the equal side or the unequal side
For Equal Side: Ways to sit = 4 (Any 1 of the 4 chairs)
For the unequal side: Ways to sit: 3 (Any 1 of the 3 chairs)
So he can sit in 7 ways.
For the rest of the people, they can be arranged in (N1)! = 10!
So, total arrangements = 7*(N1)! = 7*10!
Shortcut:
If there are N people that are needed to be arranged around an Isosceles triangle with “a” seats on each of the equal side and “b” seats on the unequal side, total arrangements = (a+b)*(N1)!