The given form to expand is (2x-4)^7
and we need to find the coefficient of x^3.
The binomial formula is
[tex](a+b)^n=\sum_{r\mathop{=}0}^nC_r^na^{n-r}b^r[/tex]where n is a positive integer and a, b are real numbers, and 0 < r ≤ n
Substituting the values a = 2x, b = -4, n = 7 and r = 4
Then we have,
[tex](2x-4)^7=C_4^7(2x)^3(-4)^4[/tex]Now the coefficient of x^3 will be
[tex]\begin{gathered} C_4^7\times8x^3\times256 \\ =\frac{7!}{4!3!}\times8\times256\times x^3 \\ =\frac{7\times6\times5}{3\times2\times1}\times8\times256\times x^3 \\ =35\times8\times256\times x^3 \\ =71680x^3 \end{gathered}[/tex]Hence, the coefficient will be 71680.
