 Hi and how are you all today? Let us proceed on with the question which is given to us. It says in a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked product A and B, 12 people liked product C and A, 14 people liked product B and C and 8 liked all the 3 products, find how many liked products C only. So let us start with our solution. By first of all, let A denote the set of people who liked product, therefore B denote the set of people who liked product B, C denote the set of people who liked product C. Now here in the question we are given that number of people who liked product A are given to us as 21, number of people who liked product B are given to us as 26, number of people who liked product C are given to us as 29 in the question. Right, now number of people who liked product A and B is given to us as 14, number of people liking product A and C are given to us as 12, whereas number of people who liked product B and C is given to us as 14. Then one more thing, number of people liking product A, B and C is given to us as 8. So can we find number of people who liked product C only? Yes we can, it will be. Now let us draw a vain diagram which will represent this information. Let this be product A, this B product B and this B product C. Right, now let us name the elements, let the elements A, B, C, D are there in set A, B, C, E, F are there in set B and G, D, C, E, B in product in set C. Right, now here number of elements A are having are A plus B plus C plus T, whereas in B we have B plus C plus E plus F whereas in C we have T plus E plus C plus G. Right, now further in A intersection B which are the elements that we are having, it is B and C which is common to both A and B right, so it is B plus C, here it is C plus E, here it is C plus T and here it is C plus E and this is represented by only one element that is common to all of them that is C. Now here by substituting the value of C in these three equation we can get the value of B, T and E. So, that will be substituting the value of C in three areas we have 8 plus T as 12, 8 plus E as 14 and 8 plus B as 14. So, let us find out the value of T that will be 4, the value of E will be 16 and the value of B will also be not 16, it will be 6 and value of B will also be 6. Now we need to find the number of people in group C only that means we need to find the numerical value of G. So, we have the value of D, we have the value of E and we have the value of C. So, substituting the value in this equation we will get the value of G and that will be our answer right. So, we have D plus C plus E plus G equals 29, let us substitute 4 plus 8 plus 6 plus G is equal to 29 that gives us the value of G as 11. So, therefore number of people who like only product C that is the value of G that is 11 right. So, this completes the entire question that was given to us. I hope you enjoyed the session and by drawing this VIN diagram and having some elements forming them into equation helped you in arriving your answer. So, I hope you understood the question and the procedure to be followed. Bye for now.