Difference between revisions of "2021 Fall AMC 12A Problems/Problem 12"
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==Solution== | ==Solution== | ||
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| + | 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> | ||
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| + | ~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 18:32, 23 November 2021
Problem
What is the number of terms with rational coefficients among the 
terms in the expansion of ![$\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?$](http://latex.artofproblemsolving.com/8/7/8/8786632e043433ca2eb557b4dff52f3f1a486b1e.png)
Solution
Each term in the expansion is of the form ![\[\binom{1000}{k}\left(x\sqrt[3]{2}\right)^k\left(y\sqrt{3}\right)^{1000-k},\]](http://latex.artofproblemsolving.com/5/1/4/5149c56d14eb7f455b7ab4656bca2d5b086b2aa2.png)
~MRENTHUSIASM
[b]APPRECIATE IF THERE'S NO EDIT. WILL FINISH UP SHORTLY[/b]
See Also
| 2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
| 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. 
