 Hello and welcome to the session. In this session we are going to discuss the following question which says that Find the probability of occurrence of the number formed with the digits 1, 2, 3, 4, 5, 6, 7, 8 such that the odd digits always occupy the odd places and the even digits always occupy the even places and the digits cannot be repeated. The number of commutations in distinct things taken all at a time is n factorial and 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 the 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. With this key idea we shall proceed with the solution. As we are given 8 digits so we have 8 places. We need to find the probability of the occurrence of the numbers formed with the digits 1, 2, 3, 4, 5, 6, 7 and 8 such that the odd digits always occupy the odd places and the even digits always occupy the even places also the digits cannot be repeated. So we have 4 even and 4 odd places as there are 4 odd places that is 1st, 3rd, 5th and 7th. So 4 digits can occupy 4 odd places such that the first place can be occupied by any of the 4 digits. So there are 4 options for this place, 3rd place can be occupied by remaining 3 odd digits. So there are 3 options for this place. Similarly 5th place will have 2 options and 7th place will have 1 option. Therefore the number of ways in which 4 odd digits can occupy 4 odd places is given by 4 into 3 into 2 into 1 that is 4 factorial which is given by 24. Similarly there are 4 even places that is 2nd, 4th, 6th and 8th and 4 even digits. Therefore the number of ways 4 even digits can occupy 4 even places is given by 4 factorial that is 24. And from the fundamental principle of multiplication we know 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 both the jobs together can be done in n into n number of ways. Similarly our first job is number of ways in which 4 digits can occupy the 4 odd places which is given by 24 and the second job would be the number of ways in which 4 even digits can occupy 4 even places which is also given by 24. Therefore by fundamental principle of multiplication we have number of favorable outcomes is given by 24 into 24 and as there are 8 digits and 8 places and know that it can be repeated and we know that the number of commutations of n testing things taken all at a time is given by n factorial. So the number of ways in which 8 digits will occupy 8 places is given by 8 factorial. Therefore total number of outcomes will be factorial therefore the required probability is given by number of favorable outcomes that is 24 into 24 upon total number of outcomes that is 8 factorial which is equal to 24 into 24 upon 8 into 7 into 6 into 5 into 4 into 3 into 2 into 1 which is equal to 1 upon 70. Therefore the required probability is given by 1 by 70 which is the final answer. This completes our session. Hope you enjoyed this session.