 Hello and welcome to the session. In this session, we will discuss a program which says in how many ways 15 flags out of which 6 are alike can be arranged in a circular order. Now our question says that there are 15 flags which are to be arranged in a circular order and out of these 15 flags 6 are alike. Then we have to find the number of ways for 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 arrange n number of objects in a circular manner, then we fix the position of one object and we are remaining n-1 objects we 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. This is the key idea we shall be using in this question. Now moving on to the solution, we have to find the circular arrangement for 15 flags out of which 6 are alike. Let the required number of arrangements be x. Let us first consider only one arrangement out of these x arrangements and we are given out of 15 flags 6 are alike. Then in this particular arrangement, 6 alike flags each other, they remain 15 minus 6 that is 9 flags. Now these 6 different flags can be arranged in 6 factorial number of ways. We have made this change in one arrangement. If this change is made to of the x number of arrangements, then the number of arrangements in which are different is given by x into 6 factorial. That is if one arrangement give rise to 6 factorial ways, then x number of arrangements will give x into 6 factorial number of ways. Now we have 15 flags in all which are to be arranged in a circular order. We now use our key idea that states that the number of ways in which n different things can form a ring is given by n-1 factorial. Here we have 15 flags in all that is n is 15 in this question. So if we fix the position of one flag, then the remaining 14 flags can be arranged in 13 factorial number of ways. So we can write 15 different flags and we arranged in a circular way in 15 minus 1 factorial ways that is 14 factorial ways. Now we have got the total number of arrangements in which all the 6 flags are different is given by x into 6 factorial and total number of arrangements in which all 15 flags can be arranged in a circular order is given by 14 factorial number of ways which is another same thing that is x into 6 factorial and 14 factorial arrangements represent the arrangement of the same set. Therefore we can write x into 6 factorial equal to 14 factorial. In this way we can get the value of x as 14 factorial divided by 6 factorial and value of x represents the required number of arrangements. Hence we say the number of ways 15 flags can be arranged out of which x are alike is given by 14 factorial divided by 6 factorial which is our final answer. This completes the session. Hope you have understood it well.