 Hi and welcome to the session. I'm Asha and I'm going to help you with the following question that says, find the coefficient of x raised to the power 5 in the product of 1 plus 2x raised to the power 6 into 1 minus x raised to the power 7 using binomial theorem. Let us begin with the solution and we have to expand 1 plus 2x raised to the power 6 into 1 minus x raised to the power 7. Now first let us open 1 plus 2x raised to the power 6 with the help of binomial theorem. We have 6C0, 1 raised to the power 6 into 2x raised to the power 0 plus 6C1, 1 raised to the power 5 into 2x raised to the power 1 plus 6C2, 1 raised to the power 4 into 2X raised to the power 2 plus 6C3 one raised to the power 3 into 2X raised to the power 3 plus 6C4 one raised to the power 2 into 2X raised to the power 4 plus 6C5 one 1 raised to the power 1 into 2x raised to the power 5 plus 6c6 1 raised to the power 0 into 2x raised to the power 6 into, now we will expand 1 minus x raised to the power 7 with the help of binomial theorem, so this is 1 raised to the power 7 plus 7c1 into 1 raised to the power 6 into minus x raised to the power 1 plus 7c2 1 raised to the power 5 into minus x raised to the power 2 plus 7c3 1 raised to the power 4 into minus x raised to the power 3 plus 7 C4 or 1 raised to the power 3up radial minus x raised to the power 4, plus 7 C5, 1 raised to the power 2up radial minus x raised to the power 5, plus 7 C6 180 1 raised to the power of 1 into minus x raised to the power of 6 plus 7c7 1 raised to the power of 0 into minus x raised to the power of 7. So we expanded these two terms with the help of binomial theorem, which can further be written as 6c0 is 1 and 1 into 1 is 1 2x raised to the power of 0 is 1 plus 6c1 is 6 into 2x 6c2 is 15 and 2x whole square is 4x square plus 6c3 is 20 and 2x whole cube is 8x cube 6c4 is 15 and 2x raised to the power 4 is 16x raised to the power 4 plus 6c5 is 6 and 2x raised to the power 5 is 32 x raised to the power 5 plus 6c6 is 1 and 1 into 1 is 1 and 2x raised to the power 6 is 64x raised to the power 6 into this is 1 raised to the power 7 since we have 7c0 1 raised to the power 7 into minus x raised to the power 0 whose value is 1 plus 7c1 is 7 into minus x plus 21 into x square since 7c2 is 21 similarly on expanding we have 35 into minus x cube plus 35 into x raised to the power 4 plus 21 into x raised to the power 5 with minus sign plus 7 x raised to the power 6 minus x raised to the power 7 which in further simplifying can be written as 1 plus 12x plus 60x square plus 160x cube plus 240x raised to the power 4 plus 192x raised to the power 5 plus 64x raised to the power 6 into 1 minus 7x plus 21x square minus 35x cube plus 35x raised to the power 4 minus 21x raised to the power 5 plus 7x raised to the power 6 minus x raised to the power 7 we have to find the coefficient of x raised to the power 5 which will be equal to now we have 2 brackets 1 we get on expanding 1 plus 2x raised to the power 6 and other we get when we expand it 1 minus x raised to the power 7 so on multiplying 1 with minus 21x raised to the power 5 we get coefficient of x raised to the power 5 that is minus 21 now multiplying 12x with 35x raised to the power 4 we will again get a term with literal x raised to the power 5 so we have plus 12 into 35 and rest when we will multiply 12x with the other terms we will get the terms which will have different literal from x raised to the power 5 similarly on multiplying 60x square with minus 35x cube we will get a term whose literal will be x raised to the power 5 so we have plus 60 into minus 35 again on multiplying 160x cube with 21x square we will get a term with x raised to the power 5 so we have next term as 160 into 21 now on multiplying 240x raised to the power 4 with minus 7x we get plus 240 into minus 7 plus now lastly on multiplying 192x raised to the power 5 with 1 we again get a term with x raised to the power 5 so lastly we have 192 into 1 which is further equal to minus 21 plus 420 minus 2100 plus 3360 minus 1680 plus 192 which is further equal to on adding all the positive integers we have 3972 minus on adding these three negative integers we have minus 3801 which is further equal to 171 so our answer is coefficient of x raised to the power 5 is equal to 171 so this completes the session take care and have a good day