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Difference between revisions of "1964 AHSME Problems/Problem 20"

(Created page with "== Problem 20== The sum of the numerical coefficients of all the terms in the expansion of <math>(x-2y)^{18}</math> is: <math>\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \te...")
 
(→‎Solution)
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==Solution==
 
==Solution==
  
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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 <math>x^0y^0</math>) is found by setting all variables equal to <math>1</math>.  Note that we don't have to worry about whether a constant is a coefficient of an "invisible <math>x^0y^0" term, because there is no such term here.
+
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 <math>x^0y^0</math>) is found by setting all variables equal to <math>1</math>.  Note that we don't have to worry about whether a constant is a coefficient of an "invisible <math>x^0y^0</math>" term, because there is no such term here.
  
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Setting </math>x=y=1<math> gives </math>(-1)^{18}<math>, which is equal to </math>1<math>, which is answer </math>\boxed{\textbf{(B)}}$.
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Setting <math>x=y=1</math> gives <math>(-1)^{18}</math>, which is equal to <math>1</math>, which is answer <math>\boxed{\textbf{(B)}}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 21:29, 23 July 2019

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 $\boxed{\textbf{(B)}}$.

See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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