1964 AHSME Problems/Problem 20

Problem 20

The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ -1\qquad \textbf{(E)}\ -19$

Solution

For any polynomial, even a polynomial with more than one variable, the sum of all the coefficients (including the constant, which is the coefficient of $x^0y^0$ ) is found by setting all variables equal to $1$ . Note that we don't have to worry about whether a constant is a coefficient of an "invisible" $x^0y^0$ term, because there is no such term here.

Setting $x=y=1$ gives $(-1)^{18}$ , which is equal to $1$ , which is answer $\boxed{\textbf{(B)}}$ .

1964 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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1964 AHSME Problems/Problem 20

Problem 20

Solution

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