2021 Fall AMC 12A Problem 12

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.