 Hello and welcome to the session. In this session we discuss a problem based on circular permutation which says in how many ways 15 delegates can be seated around a circular table so that all shall not have the same neighbors in any two arrangements. Let's have a look at the question. There are 15 delegates who are to be seated around a circular table with a condition that all should not have the same neighbors in any of the arrangements. We know that the number of ways in which n persons can sum a ring is given by n minus 1 factorial that is if we have to make circular arrangement for n number of persons then we fix the position of 1 person and the remaining n minus 1 number of persons can be arranged in n minus 1 factorial number of ways or we can say that the number of ways in which n persons can sum a ring is given by n minus 1 factorial also we know that number of ways when clockwise and t clockwise arrangements not different is given by n minus 1 factorial divided by 2 that is when clockwise and t clockwise arrangements are not different then the number of ways they are finding this arrangement is given by n minus 1 factorial divided by 2. This is the key idea we shall be using in this question. Let's move on to the solution as there are 15 delegates who are to be seated around a circular table then by using the key idea it says the number of ways in which n person can sum a ring is given by n minus 1 factorial and we have 15 delegates to be seated around a circular table that is here the value of n is equals to 15 and if we fix the position of 1 delegate then the remaining 14 delegates can be placed in 13 factorial number of ways so we can write the number of ways in which 15 delegates will sum a ring is given by 15 minus 1 factorial which is equal to 13 factorial after being placed in this manner each delegate will have the same label in clockwise and anti-clockwise arrangement and by using the key idea you have the number of ways when clockwise and anti-clockwise arrangements are not different is given by n minus 1 factorial divided by 2 so we get the required number of arrangements given by half multiplied by 13 factorial which is equal to multiplied by 13 factorial can be written as 14 multiplied by 13 factorial which on cancellation gives 7 multiplied by 13 factorial therefore number of ways in which 15 delegates will be seated around a circular table shall not have the same neighbors in any two arrangements 7 multiplied by 13 factorial which is our final answer this completes our session hope you have understood it well