1965 AHSME Problems/Problem 14

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Problem 14

The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is:

$\textbf{(A)}\ 0 \qquad  \textbf{(B) }\ 7 \qquad  \textbf{(C) }\ 14 \qquad  \textbf{(D) }\ 128 \qquad  \textbf{(E) }\ 128^2$

Solution

Notice that the given equation, $(x^2 - 2xy + y^2)^7$ can be factored into $(x-y)^{14} = \sum_{k=0}^{14} \binom{14}{k}(-1)^k\cdot x^{14-k}\cdot y^k$.

Notice that if we plug in $x = y = 1$, then we are simply left with the sum of the coefficients from each term. Therefore, the sum of the coefficients is $(1-1)^7 = 0, \boxed{\textbf{(A)}}$.

See Also

1965 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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