Problem #1689

 1689 A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? $\textbf{(A)}\ \dfrac{49}{512} \qquad\textbf{(B)}\ \dfrac{7}{64} \qquad\textbf{(C)}\ \dfrac{121}{1024} \qquad\textbf{(D)}\ \dfrac{81}{512} \qquad\textbf{(E)}\ \dfrac{9}{32}$ This problem is copyrighted by the American Mathematics Competitions.
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• Reduce fractions to lowest terms and enter in the form 7/9.
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