 Hi and how are you all today? I'm Priyanka and let us discuss this question. It says find the middle term in the expansion of x by 3 plus 9y the whole raise to the power 10. But before proceeding on with the solution, we should be well versed that if the n given to us is even, then we can find the number of terms by adding 1 to n and since n is even n plus 1 will be odd and therefore the middle term is obtained by dividing n by 2 and adding 1 to it and it will become n by 2 plus 1th term which will be the middle term if n is even. The knowledge about the middle terms are the key ideas to these questions. Let us proceed on with our solution. Now here the expression given to us is x by 3 plus 9y the whole raise to the power 10. Here n is equal to 10 that is even, right? So middle term can be found out by dividing 10 by 2 and then adding 1 to it. That will be the 6th term. So let us first find out the 6th term that will be t6 that can be written as t5 plus 1 where r will become 5. Now let us substitute. Now here x by 3 is our a and b is 9y and n is 10. So we have 10c5 a that is x by 3 raise to the power 10 minus 5 b that is 9y raise to the power 5. On simplifying we have 10 factorial divided by 5 factorial multiplied by 5 factorial multiplied by x raise to the power 5 divided by 3 raise to the power 5 multiplied by 9 raise to the power 5 multiplied by y raise to the power 5. Further 10 factorial can be written as 10 into 9 into 8 into 7 into 6 into 5 factorial divided by 5 factorial and the other 5 factorial can be written as 5 into 4 into 3 into 2 into 1. Further x raise to the power 5 divided by 3 raise to the power 5 means 3 times 5 5 times 3 multiplied by 9 raise to the power 5 multiplied by y raise to the power 5. Further simplifying we have left with 9 into 4 into 7 into 81 into 3 into x raise to the power 5 y raise to the power 5 which can further be written as 61236 x raise to the power 5 y raise to the power 5 which is the answer that is the sixth term as well as the middle term of x by 3 plus 9y raise to the power 10. 61236 x raise to the power 5 y raise to the power 5 is the required answer. This ends the session. Bye for now.