The term independent of x in the expansion of (160 - x^881) · (2x^2 - 3x^2)^6 is equal to

Correct option is D. −36
Step 1:- Find the general term
Given, (160−x^881)⋅(2x^2−3x^2)^6
We know that, The (r+1)th term of the binomial expansion (a+b)^n = nCr a^(n-r) b^r
The (r+1)th term of the binomial expansion (2x^2−3x^2)^6= 6Cr (2x^2)^(6-r) (-3x^2)^r = 6Cr (-1)^r 2^(6-r) 3^r x^(12-2r+2r) = 6Cr (-1)^r 2^(6-r) 3^r x^12
Step 2 :- Find the independent term
The term independent of x in (160 - x^881) · (2x^2 - 3x^2)^6 will only come from the constant term in (2x^2 - 3x^2)^6 multiplied by 160 because the other term in the expression contains x^881. Thus we will look for the term without x in (2x^2 - 3x^2)^6 = (-x^2)^6 = x^12
For independent of x, the power of x must be 0. Since we have x^12 this means there is no term independent of x from this expansion. Therefore we must consider the constant term in the expansion of (2x^2 - 3x^2)^6. This is given by:
(2x^2 - 3x^2)^6 = (-x^2)^6 = x^12
There is no term independent of x in this expansion.
However, the question may have a typo and the intended expression might have been different. Let's assume there was a typo in the question and that the expression intended was (160 - x^881)(2x - 3x)^6 = (160 - x^881)(-x)^6 = (160 - x^881)x^6
In this case, the term independent of x would be obtained when x^6 is multiplied by 160. This will result in a constant term independent of x.
Let's consider another possible interpretation. The expression might be (160 - x^881)(2x^2 - 3x)^6. In this case, we need to find the term independent of x from (2x^2 - 3x)^6. The general term in the binomial expansion is given by:
6Cr (2x^2)^(6-r) (-3x)^r = 6Cr 2^(6-r) (-3)^r x^(12 - 2r + r) = 6Cr 2^(6-r) (-3)^r x^(12-r)
For the term to be independent of x, we need 12 - r = 0, which means r = 12. However, this is not possible since r must be between 0 and 6. Therefore, there is no term independent of x in this expansion either.
Let's assume the question intended (160 - x^881)(2x - 3x)^6 = (160 - x^881)(-x)^6 = (160 - x^881)x^6. In this case the term independent of x is 160. This is not one of the options.
The provided solution seems to have errors or is based on an incorrectly stated question.
Hence, there seems to be an error in the problem statement or the solution provided.