 Hello and welcome to the session. In this session we discuss a problem based on circular commutation which says in our group discussion 7 boys and 7 girls are to be seated around a circular table so that no 2 boys will sit together. In how many ways we can do this? Let's understand the question first. In this question there are 7 boys and 7 girls who are to be seated around a circular table with a condition that no 2 boys will be seated together. And we have to find the number of ways for this arrangement. As we already know the number of ways in which in persons can form a ring is given by n minus 1 factorial. That is if you have to make circular arrangements for n number of objects then we fix the position of any one of the object and the remaining n minus 1 number of objects can be arranged in n minus 1 factorial number of ways. Hence we can say the number of ways in which n persons can form a ring is given by n minus 1 factorial. Also we have one more principle known as fundamental principle of multiplication says that if there are 2 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 where the jobs together can be done in n multiplied by n number of ways. This is the key idea we shall be using in this question. Let's move on to the solution. We are given that there are 7 boys and 7 girls to be seated around a circular table. Let us assume that first set up their seats and we have to arrange the position of these 7 girls in a circular manner. Using the key idea we know that n persons can form a ring in n minus 1 factorial number of ways. If there are 7 ways in all then using the key idea we can fix the position of one girl then the remaining 6 girls can be placed in 6 factorial number of ways. Or we can say that the number of ways 7 girls can be seated is given by 7 minus 1 factorial which is equal to 6 factorial. When all the girls have been seated then there remain 7 places of the boys each between 2 girls and these 7 places can be filled in 7 factorial number of ways. So the number of ways in which boys can sit is equal to factorial. Now there are 6 factorial ways in which 7 girls can be seated and 7 factorial ways in which 7 boys can be seated. Now we have to find out the number of ways in which 7 boys and 7 girls will be seated together. Now using the fundamental principle of multiplication as given in the key idea it says that if there are 2 jobs to be done such that the first set 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 multiplied by n number of ways. So we have our first job that is the number of ways in which girls can be seated is given by 6 factorial number of ways Our second job that is number of ways in which boys can be seated which is given by 7 factorial. So by using the same principle we get the number of ways in which 7 boys and 7 girls will be seated around a circular table is equal to 6 factorial multiplied by 7 factorial. Now 6 factorial can be written as 6 into 5 into 4 into 3 into 2 into 1 which is 7 multiplied by 7 factorial which gives 7 into 6 into 5 into 4 into 3 into 2 into 1 which on multiplication gives 3,628,800. Hence the number of ways 7 boys and 7 girls can be seated so that no 2 boys are adjacent is 3,628,800 which is our final answer. This completes this session hope you have understood it well.