 Hello and welcome to the session. I am Asha and I am going to help you with the following question that says out of 100 students 2 sections of 40 and 60 are formed if you and your friend are among the 100 students what is the probability that A you both enter the same section B you both enter the different section. Let us now start with the solution. Select the two sections A and B and the number of students in section A before B and in section B B 60. Now the first part is we have to find the probability that you and your friend both enter the same section. Let us suppose let you both enter section A. Now section A has 40 students if you both enter in section A then 38 students selected out of 98 students which we get. I am subtracting 2 from 100 since 2 students are already in section A you and your friend so 38 students are selected out of 98 students in 98 say 38 ways. Let us denote the event X by the fact that you and your friend both enter section A. Now the total possible number of choices of 40 students out of 100 as 100 C 40 therefore probability of X that you both enter the same section is given by the number of favorable events that is 98 C 38 upon the total number of possible choices that is 100 C 40. Now simplifying it further 98 C 38 can be written as 98 factorial upon 38 factorial into 98 minus 38 factorial into 40 factorial into 100 minus 40 factorial upon 100 factorial. This is further equal to 98 factorial upon 38 factorial into 60 factorial into 40 factorial can be written as 40 into 99 into 38 factorial into 60 factorial upon 100 factorial can be written as 100 into 99 into 98 factorial. Now canceling 98 factorial with 98 factorial 38 factorial with 38 factorial and 60 factorial with 60 factorial 2 2 0 4 2 5 is a 10 3 13's are 39 and 3 into 33 yes 99 so we have 26 upon 165 therefore probability that you both enter section A is equal to 26 upon 165. Now let us find the probability that both enter section B. Now number of students in section B is equal to 16 suppose you and your friend both enter section B so now we have to choose 58 students from 98 students so we have to select 58 students from 98 students and the number of favorable choices that is the number of favorable events of choosing 58 students from 98 students is 98 C 58 and again as we have done earlier total number of possible choices is equal to 100 C 60 in this case since we have to select 60 students out of 100 and let us denote that both enter section B by the event y so we can say that probability of y is equal to number of favorable choices that is 98 C 58 upon the total number of possible choices 100 C 60 which can further be written as 98 factorial upon 58 factorial into 98 minus 58 factorial into 60 factorial into 100 minus 60 factorial upon 100 factorial so we have 98 factorial upon 58 factorial into on subtracting we have 40 factorial into 60 factorial can be written as 60 into 59 into 58 factorial and 100 minus 60 factorial is 40 factorial upon 100 factorial can be written as 100 into 99 into 98 factorial now cancelling 98 factorial cancels up with 98 factorial 40 factorial with 40 factorial 58 factorial with 58 factorial and we further have 2 3's are 6 2 5's are 10 and 3 1's are 3 3 into 33 is 99 so we have 59 upon 165 therefore probability both enter section B is equal to 59 upon 165 now probability of x or y that is you both enter either section A or section B is given by probability of x plus probability of y and probability of x is 26 upon 165 plus probability of y is 59 upon 165 which is equal to 85 upon 165 and now 5 into 17 is 85 5 into 33 is 165 so we have 17 upon 33 so answer to the first part that is probability that you and your friend both enter same section is equal to 17 upon 33 so this completes the first part now let's proceed on to the second part here we have to find probability that you both enter the different sections that is probability of x or y complement which is equal to 1 minus probability of x or y that is 1 minus 17 upon 33 which is equal to 16 upon 33 so answer to the second part is probability that you both enter different sections is equal to 16 upon 33 so this completes the session hope you have understood it bye and take care