Seating Arrangement (Part-1)

Q:If there are 3 chairs on each side of square, then what are the total number of ways in which we can arrange 12 people around the square table?

Ans: Sides of square are equal so a person X can sit around table in 3 ways(Any of the 3 chairs can be put on one side) 

So X can sit either on left chair, in the middle or the right chair. We can arrange other people in 

(N-1)!  i.e. (12-1)!=11!

Hence ways to arrange 12 people in=

3*(12-1)!=3*11! ways

Shortcut: If there are N people to be assigned a square table with  (N/4) seats on one side of it

Total ways=(N/4)*(N-1)!

Q: In how many ways can we arrange 10 people around rectangular table such that there are 3 chairs on longer side and 2 on shorter side? 

Ans. So if we consider the first person X

X can either sit on longer side or shorter side. 

For longer side there are 3 possibilities =left, right and center

For shorter side there are 2 possibilities =left and right

Hence X can sit in 5 ways

Total arrangement =5*(10-1)!

                                      =5*9!

Shortcut:

If we have N people that are needed to be arranged around rectangular table with 'a' seats on the longer side and 'b' seats on shorter side. 

Total ways: (a+b) *(N-1)!

Q:There are 12 people needed to be arranged around an equilateral triangular table, which has 4 seats on each side. How many arrangements are possible? 

Ans:

Equilateral triangle has all sides equal so first person i.e. X may sit on any one side

Ways to sit : 4(any of 4 chairs) 

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

Shortcut:

N people are arranged around equilateral trangle

Total arrangements:(N/3)*(N-1)!

Q: 11 people arranged around isosceles triangle which has 4 seats each on equal sides and 3 seats on other. How many arrangements are possible? 

Ans:

So  first person.i.e X can either sit on equal side or the unequal sides

For unequal side=3 ways

For equal side=4 ways

So total ways= 7*(11-1)!=7*(10)!

Shortcut:

N people that are around isosceles triangle with 'a' seats on each of equal side and 'b' seats on unequal side

=(a+b) *(N-1)!