Difference between revisions of "2021 Fall AMC 12A Problems/Problem 12"

Line 7: Line 7:
 
==Solution==
 
==Solution==
  
Each term in the expansion is of the form <cmath>\binom{1000}{k}\left(x\sqrt[3]{2}\right)^k\left(y\sqrt{3}\right)^{1000-k},</cmath>
+
By the Binomial Theorem, each term in the expansion is of the form <cmath>\binom{1000}{k}\left(x\sqrt[3]{2}\right)^k\left(y\sqrt{3}\right)^{1000-k}=\binom{1000}{k}2^{\frac k3}3^{\frac{1000-k}{2}}x^k y^{1000-k},</cmath> where <math>k\in\{0,1,2,\ldots,1000\}.</math>
 +
 
 +
This problem is equivalent to counting the values of <math>k</math> such that both <math>\frac k3</math> and <math>\frac{1000-k}{2}</math> are integers. Note that <math>k</math> must be a multiple of <math>3</math> and a multiple of <math>2,</math> so <math>k</math> must be a multiple of <math>6.</math> There are <math>\boxed{\textbf{(C)}\ 167}</math> such values of <math>k:</math> <cmath>6\cdot0, 6\cdot1, 6\cdot2, \ldots, 6\cdot166.</cmath>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
 
[b]APPRECIATE IF THERE'S NO EDIT. WILL FINISH UP SHORTLY[/b]
 
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2021 Fall|ab=A|num-b=11|num-a=13}}
 
{{AMC12 box|year=2021 Fall|ab=A|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:48, 23 November 2021

Problem

What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\ 500 \qquad\textbf{(E)}\ 501$

Solution

By the Binomial Theorem, each term in the expansion is of the form \[\binom{1000}{k}\left(x\sqrt[3]{2}\right)^k\left(y\sqrt{3}\right)^{1000-k}=\binom{1000}{k}2^{\frac k3}3^{\frac{1000-k}{2}}x^k y^{1000-k},\] where $k\in\{0,1,2,\ldots,1000\}.$

This problem is equivalent to counting the values of $k$ such that both $\frac k3$ and $\frac{1000-k}{2}$ are integers. Note that $k$ must be a multiple of $3$ and a multiple of $2,$ so $k$ must be a multiple of $6.$ There are $\boxed{\textbf{(C)}\ 167}$ such values of $k:$ \[6\cdot0, 6\cdot1, 6\cdot2, \ldots, 6\cdot166.\]

~MRENTHUSIASM

See Also

2021 Fall AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png