The coefficient of x^7 in the expansion of ((x^2)/(2)-2x)^8 is:

1960 AMC 12 Problem 20

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20.

The coefficient of

x^7

in the expansion of

\left(\frac{x^2}{2}-\frac2x\right)^8

is:

56

-56

14

-14

0

In a term using

k

copies of

-\frac{2}{x},

determine the exponent of

x

Solve

16-3k=7

and then compute that binomial coefficient

The term using

k

factors of

-\frac{2}{x}

has exponent

2(8-k)-k=16-3k.

To obtain

x^7,

we need

k=3.

Its coefficient is

\binom83\left(\frac12\right)^5(-2)^3=-14.

Thus, the correct answer is D.