 Hello and welcome to the session. In this session we will discuss a problem which says, in how many ways 12 different things can be arranged around a circular table so that two particular things are always together. In this question we are given 12 different things to be arranged around a circular table with a condition that two particular things are always together. Then we have to find out in how many ways we can make this arrangement. We know that the number of ways in which n different things can form a ring is given by n-1 factorial. That is, if we have to make circular arrangement for n number of objects then we fix the position of one object and the remaining n-1 objects can be arranged in n-1 factorial number of ways or we can say that the number of ways in which n different things can form a ring is given by n-1 factorial. We also know one more principle known as fundamental principle of multiplication which states that if there are two jobs to be done such that the first job can be done in n number of ways and second job can be done in n number of ways then both the jobs together can be done in n into n number of ways. This is the key idea we shall be using in this question. Let's move on to the solution. Now in the question we have 12 different things to be arranged around a circular table. Also we have one more condition that the two particular things are always together in all the arrangements. There are 12 different things in all to be arranged in a circular order and we know that the two particular things are always together. So let us assume t1 and t2 be the two particular things. Now considering t1 and t2 as one thing then out of 12 things we are left with 11 things. Now we have 11 things in all. Now for the circular arrangement of these 11 things we can use the key idea which states that the number of ways in which n different things can form a ring is given by n minus 1 factorial that is if we have 11 objects in all then we can fix the position of one object and the remaining 10 objects can be arranged in 10 factorial number of ways. Therefore we can say that the number of ways in which 11 things can be arranged around a circular table is given by 11 minus 1 factorial that is 10 factorial. Also the two particular things can arrange themselves to a factorial number of ways that is either as t1, t2 or t2, t1. Now we know that 11 things can be arranged around a circular table in 10 factorial number of ways and the two particular things can be arranged in two factorial number of ways and we have to find out the arrangement for 12 different things. From the key idea we have fundamental principle of multiplication which states that if there are two jobs to be done such that the first job can be done in n number of ways and second job can be done in n number of ways then both the jobs together can be done in n into n number of ways. Now in this question our first job is the arrangement of 11 things in the circular order which is given by 10 factorial number of ways and our second job is the arrangement of the two particular things which is given by two factorial number of ways. So by using the fundamental principle of multiplication we can say that therefore the total number of ways in which all the 12 things will be arranged is equal to 10 factorial multiplied by 2 factorial which is equal to 10 factorial multiplied by 2 multiplied by 1 which gives 2 multiplied by 10 factorial. Hence we can say that the total number of ways in which 12 different things can be arranged around a circular table such that two particular things are always together is given by 2 multiplied by 10 factorial which is our final answer. This completes the session. Hope you have understood it well.