AS level Binomial expansion help please?

I agree with your answer to the first part.

However I think you have misread the second part of the question.

I think it should read

"Find the coefficient of x in the expansion of (2 - x)(2x - 1)^6".

On this assumption we would have

(2 - x )(1 - 12x + 60x² - 160x³ +...)

Just looking at possible coefficients for x we have (2 x -12) + (-1) x 1 = -25

the 1st few words of the binomial growth of (2k + x)? are :- (2k)? + n(2k)??¹x + n(n - a million)/2! *(2k)??²x² + n(n - a million)(n - 2)/3! * (2k)??³x³ +... a) because of the fact the coefficients of x² and x³ are an identical then :- n(n - a million)/2! *(2k)??² = n(n - a million)(n - 2)/3! * (2k)??³ 2k =(n - 2)/3 : 6k = n - 2 : n = 6k + 2 b) As n = 6k + 2 then whilst ok = ?, n = 6.....and using the faster growth provides :- (2*?)^6 + 6*(2*?)^5.x + 6*5/2! *(2*?)^4.x² + 6*5*4/3! * (2*?)³.x³ = (4096/729) + (2048/80 one)x + (1280/27)x² + (1280/27)x³........that's what you may anticipate because of the fact the coefficients of x² and x³ are equivalent.

Chicken little.