1974 AMC 12 Problem 3

Attempt Problem 3 of the 1974 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 1974 AMC 12 solutions, or check the answer key.

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

The coefficient of

x^7

in the polynomial expansion of

(1+2x-x^2)^4

-8

12

6

-12

none of these

A degree-

7

term must select degrees

2,

2,

2,

1

from the four factors

Choose which factor contributes

2x

; the other three contribute

-x^2

The only way four selected terms can have total degree

7

is to choose

2x

once and

-x^2

three times. There are

4

choices for the first factor, so the coefficient is

4(2)(-1)^3=-8.

Therefore, the correct answer is A.

Problem 3 in other years