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One Thousand and One Terms

Source: AMC 12A #12

November 11, 2021

Problem Statement

What is the number of terms with rational coefficients among the 10011001 terms of the expression (x23+y3)1000( x \sqrt[3]{2} + y \sqrt{3})^{1000}?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 166<spanclass=′latex−bold′>(C)</span> 167<spanclass=′latex−bold′>(D)</span> 500<spanclass=′latex−bold′>(E)</span> 501<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 166 \qquad<span class='latex-bold'>(C)</span>\ 167 \qquad<span class='latex-bold'>(D)</span>\ 500 \qquad<span class='latex-bold'>(E)</span>\ 501
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