 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is consider a binary operation star on the set 1, 2, 3, 4, 5 given by following multiplication table. Now first part is compute 2 star 3 star 4 and 2 star bracket star 3 star 4. Second part is star commutative. Third part is compute 2 star 3 star 4 star 5. First of all let us understand the key idea to solve the given question. Given any operation table having n rows and n columns with each entry being an element of set A equal to A1 comma A2 comma till An we can define a binary operation star from A cross A to A given by Ai star Aj which is equal to entry in the i-th row and jth column of the operation table. Let us start with the solution now. First of all we have to calculate bracket 2 star 3 star 4. Now we can see from table 2 star 3 is equal to 1 so we will substitute the value of 2 star 3 equal to 1 and get 1 star 4. Now we can see from the table 1 star 4 is again equal to 1 so we will substitute the value of 1 star 4 equal to 1 so we get 2 star 3 star 4 is equal to 1. Let us now calculate 2 star bracket 3 star 4 below 3 star 4 is equal to 1 so we will substitute the value of 3 star 4 equal to 1 and get 2 star 1 from table again we can see 2 star 1 is equal to 1 so we will substitute the value of 2 star 1 equal to 1 and get 2 star bracket 3 star 4 equal to 1. Now this completes the first part of the solution right. Now let us see for the second part. Second part is A star commutative we know that in the question itself the table given to us is multiplication table. So we find that A star A must be equal to A and A star B is equal to B star A for the multiplication table. As we see it clearly in the table that it is the table is symmetrical about the diagonal starting from upper left corner and ending at the lower right corner. So star is commutative. Now this completes the second part. Now third part is we have to compute 2 star 3 star 4 star 5 rewrite the question 2 star 3 star 4 star 5. Now we will find the values of 2 star 3 and 4 star 5 from the table. We can see from the table 2 star 3 is equal to 1 and 4 star 5 is also equal to 1. So we can write 2 star 3 star 4 star 5 is equal to 1 star 1. This is because 2 star 3 is equal to 1 and 4 star 5 is also equal to 1. Now we will again see the table. We can see 1 star 1 is also equal to 1. So we will substitute the value of 1 star 1 equal to 1. So we get 2 star 3 star 4 star 5 is equal to 1. So the required answer for the first part is 2 star 3 star 4 is equal to 1 and 2 star bracket 3 star 4 is also equal to 1. Answer for the second part is yes star is commutative and answer for third part is 2 star 3 star 4 star 5 is equal to 1. This is our required answer. Hope you understood the session. Take care and goodbye.