1965 AHSME Problems/Problem 14

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 $(1-1)^2 = 0, \boxed{\textbf{(A)}}$

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1965 AHSME Problems/Problem 14

Problem 14

Solution