 Hello and welcome to the session. In this session, we are going to discuss circular commutations. Till now we have discussed arrangement of objects in a row. For example, if we arrange 5 numbers that is 1, 2, 3, 4, 5, the two of the arrangements 1, 2, 3, 4, 5 and if 1, 2, 3, 4 are two different arrangements in a row and this type of commutations is known as linear commutations that if these arrangements are written along a closed curve that is a circle then the above two discussed arrangements are same. Such type of commutations is called circular commutations. We can see that circular arrangements are different only when the relative order of the objects is changed otherwise they are same as the number of circular commutation depends on the relative position of the objects. Therefore, here we consider one object as fixed and the remaining objects are arranged in case of linear arrangements. For example, if we have to arrange n different objects in a circular way then we fix the position of one object and then arrange the remaining n-1 objects in all possible ways and this can be done in n-1 factorial ways. Let a1, a2 up to an the n distinct objects and one of the ways in which they can form a ring is shown. The figure shows one of the ways in which n distinct objects can be arranged. Now, starting with different letters and considering them in clockwise direction the various arrangement of these objects will be obtained as a1, a2, up to an-1, an. Now, starting with a2 we have a2, a3, up to an-1, an, then a1. Starting with a3 we have a3, a4, up to an-1, an, a1, a2. Now, starting with a4 we have a4, a5, up to an-1, then an, a1, a2, a3. Now, proceeding in the same manner, lastly we get starting with an, a1, a2, up to an-1. Clearly, the circular permutation provides n linear permutations, hence if the required number of circular arrangements of the n objects be x, one circular permutation provides n linear permutations, therefore the total number of linear permutations will be x into n that is xn. We also know that the total number of linear arrangements of n distinct objects is given by n factorial. Therefore, we have xn is equal to n factorial which implies that x is equal to n factorial by n that is equal to n factorial can be written as n into n minus 1 factorial upon n which is equal to n minus 1 factorial. Therefore, the number of ways in which n objects can form a ring is given by n minus 1 factorial. In the above discussion we have considered anti-clockwise and clockwise order arrangements as distinct but in actual practice we have two types of circular permutations. One, they are counterclockwise and clockwise arrangements are distinguishable. For example, seating persons around the circular table, this is the clockwise arrangement of objects and this is the anti-clockwise arrangement of objects. Here we can see that the clockwise and anti-clockwise arrangements are distinguishable. And second is they are counterclockwise and clockwise arrangements are not distinguishable. For example, arrangement of beads in the necklace. Now, if we arrange the objects in this clockwise and anti-clockwise manner then their arrangements would be same that is not distinguishable. Now we shall discuss some important points in circular permutations. First we have number of circular arrangements and different things are n minus 1 factorial when clockwise and anti-clockwise arrangements are not different Then the number of circular arrangements and different things are given by 1 by 2 into n minus 1 factorial. Next we have number of circular arrangements and different things taken are at a time when clockwise and anti-clockwise arrangements are taken as different is given by npr by r or can also be written as ncr into r minus 1 factorial that is ways of selecting r elements from n and arranging them in a circular permutation. Next we have number of circular arrangements of n different things taken are at a time when clockwise and anti-clockwise arrangements are not different is given by npr upon 2r or can also be written as ncr into r minus 1 factorial by 2 that is ways of selecting r elements from n and arranging them in a circular permutation when clockwise and anti-clockwise arrangements are not different. For example the number of ways in which 20 different beats can be arranged to form a netlist is given by n minus 1 factorial by 2. As we know that when clockwise and anti-clockwise arrangements are not different the number of circular arrangements of n different things are 1 by 2 into n minus 1 factorial. In the arrangement of beats the clockwise and anti-clockwise arrangements are not different. Therefore it is given by n minus 1 factorial by 2 and the value of n is 20 so we have 20 minus 1 factorial by 2 which is equal to 1 by 2 into 19 factorial. Now we shall find out number of circular arrangements of 20 different beats taken 15 at a time which is given by npr upon 2 into r. Since we know that number of circular arrangements of n different things taken r at a time when clockwise and anti-clockwise arrangements are not different is given by npr upon 2r. Here the value of n is given by 20 and r is equal to 15 therefore we have 20 p15 upon 2 into 15 which is equal to 20 factorial upon 20 minus 15 factorial whole upon 2 into 15 which is equal to 20 factorial upon 20 minus 15 factorial that is 5 factorial into 2 into 15. Now 20 factorial can be written as 20 into 19 into 18 into 17 into 16 into 15 into 14 factorial upon 5 factorial can be written as 5 into 4 into 3 into 2 into 1 into 2 into 15 which is equal to 19 into 3 into 17 into 8 into 14 factorial which is equal to 7752 into 14 factorial. Number of circular arrangements of nt different beats taken at a time is given by 7752 into 14 factorial which is the required answer. This completes our session hope you enjoyed this session.