Binomial, Eh!

$f(x)=\{2 - x(1-3x) \}^6. ~~~\therefore ~f(1)=1 - 6 + 15 - 20 + 15 - 6 + 1.\\ \therefore (A - B)^6=A^6 - 6A^5*B + 15A^4*B^2 \color{#3D99F6}{ - 20*A^3*B^3 + 15*A^2*B^4 - 6*A*B^5 }+ B^6\\ No ~ term ~up ~ till~~\{x(1+3x) \}^2 ~ would ~ give ~ any ~ x^5 ~ term.\\ \{x(1+3x) \}^6 ~ also ~ would ~ not ~ give ~ any ~ x^5 ~ term.\\ \text{So there are only three terms and they add up to a - tive. So only A and C are applicable !!!}$
$The ~ solution ~ is ~ as ~ under. . \\ - 20*2^3*x^3*(1 - 9x ~~ \underline{+27x^2}~ - ...) +15*2^2*x^4*(1 ~~ \underline{- 4*3x} + ...) - 6*2*x^5(1 + ...)\\ =( - 4320 - 720 -12)x^5.$

We can use multinomial theorem to solve this question $(2-x+3x^2)^6$ = $\displaystyle \sum(6!/(a!×b!×c!))×(2^a×(-x)^b×(3x^2)^c)$ Here a+b+c=6 .We want to get the coefficient of $x^5$ so b+2c=5; so we will have three possibilities

1. a=3 , b=1 , c=2
2. a=2 , b=3 , c=1
3. a=1 , b=5 , c=0

Hence there are three terms containing $x^5$ . The coefficients in each case will be

1. -((6!/(1!×2!×3!))× $2^3$ × $3^2$ )= -4320

Similarly

1. -720
2. -12

respectively

So the sum of coefficients will be $\boxed{-5052}$ .