1965 AMC 12 Problem 14

Attempt Problem 14 of the 1965 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1965 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

The sum of the numerical coefficients in the complete expansion of

(x^2-2xy+y^2)^7

is:

0

7

14

128

128^2

Set both variables equal to

1

to obtain the coefficient sum

The expression inside the seventh power becomes

1-2+1

Setting

x=y=1

makes every monomial equal

1,

so the value is the sum of the numerical coefficients. It is

(1-2+1)^7=0.

Therefore, the correct answer is A.

Problem 14 in other years