 Hello and welcome to the session. In this session we discuss the following question which says, in a class of 60 boys there are 45 boys who play cards and 30 boys who play carrom. Find how many boys play both games, how many boys play cards only, how many boys play carrom only. Before we move on to the solution, let's discuss some cardinal properties of set. First we have number of elements in the set A union B is equal to the number of elements in the set A plus number of elements in the set B minus number of elements in the set A intersection B. Then next cardinal property is number of elements in the set A minus B is equal to number of elements in set A minus number of elements in set A intersection B. Then we have number of elements in the set B minus A is equal to number of elements in set B minus number of elements in set A intersection B. This is the key idea that we use for this question. Let's move on to the solution now. First of all we suppose let A be the set of boys who play cards and let B be the set of boys who play carrom. Then A union B would be the set of boys who play cards or carrom or both and A intersection B would be the set of boys who play cards and carrom both. A minus B would be the set of boys who play cards only and B minus A would be the set of boys who play carrom only. Now since we are supposed to find out the number of boys who play both games so this means we will find out number of elements in the set A intersection B and also we are supposed to find out how many boys play cards only. So this means we have to find out the number of elements in the set A minus B and in the third part we are supposed to find out the number of boys who play carrom only that means we will also find out number of elements in the set B minus A. Now we had considered A to be the set of boys who play cards and B to be the set of boys who play carrom and in the question we have that 45 boys play cards and 30 boys play carrom so this means the number of elements in set A is equal to 45 since 45 boys play cards and number of elements in set B is 30 since in the question it is given that 30 boys play carrom. Now A union B is the set of boys who play cards or carrom or both this means it would be the total number of boys in the class and we are given that there are total of 60 boys in the class thus we have number of elements in the set A union B is equal to 60. Now we have the cardinal property according to which we have number of elements in the set A union B is equal to number of elements in the set A plus number of elements in set B minus number of elements in set A intersection B. From here we have number of elements in set A intersection B is equal to number of elements in set A plus number of elements in set B minus number of elements in set A union B. So putting the values we get 45 plus 30 minus 60. This is the number of elements in the set A intersection B. So further we have number of elements in set A intersection B is equal to 15. Therefore for the first part we have that 15 boys play both the gains. Now next we will find out number of elements in the set A minus B that is number of boys who play cards only this would be equal to number of elements in set A minus number of elements in the set A intersection B according to the cardinal property that we have stated in the key idea. So putting the values we get this is equal to 45 which is the number of elements in the set A minus 15 that is the number of elements in the set A intersection B and so this is equal to 30 that is number of elements in the set A minus B. And therefore we say that 30 boys play cards only. Now next we are supposed to find out number of elements in the set B minus A that is the number of boys who play carom only this would be equal to number of elements in the set B minus number of elements in the set A intersection B now putting the values we get this is equal to 30 minus 15 which is equal to 15 that is number of elements in the set B minus A is equal to 15 thus we have 15 boys play carom only. So we have got the answers for all the three parts of the question for the first part we have that 15 boys play both the games then for the second part we have 30 boys play cards only and for the third part we have 15 boys play carom only. So this completes the session hope you have understood the solution of this question.