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 (N-1)!

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 (n-1)! 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 = (N-1)! = (12-1)! = 11!

Hence ways to arrange 12 different persons in the question= 3*(12-1)! = 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) * (N-1)!

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* (10-1)! = 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) * (N-1)!

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*(12-1)! = 4*11!

Shortcut :

If there are N people that are needed to be arranged around an equilateral triangle, total arrangements = (N/3) *(N-1)!

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 (N-1)! = 10!