## Problem 1160

Each of 2 countries sends 3 delegates to a negotiating conference. A rectangular table is used with 3 chairs on each long side. If each country is assigned a long side of the table (operation 1), how many seating arrangement are possible?

**Solution:-**

Number of Permutations of n Objects

The number of permutations of n distinct object without repetition is given by

P_{n,n} = n(n-1)(n-2)*……..2*1= n!

Note the operation 1 is the assignment of a country to the side of a table. There are two sides of the table, so two permutations of how the sides may be assigned to the countries .

P _{2,2}= 2

Once the side are assigned the seating arrangement of each country can be determined. The arrangements for the countries are separate. As a result, the total number of arrangements (once the sides are chosen) is the product of the number of arrangements for each country.

Arranging the seating means putting the delegates in an order, hence the solution requires permutations.

The number of ways one country’s delegates may be seated is

P _{3,3} = 3! = 6

The total number of seating arrangements is given by.

= P_{2.2}*P_{3,3}*P_{3,3} = 2*3!*3! = 2*6*6 = 72.