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black and white

Source: 2021 AMC 10A #13

November 11, 2021
AMCAMC 10AMC 10 A

Problem Statement

Each of 66 balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other 55 balls?
<spanclass=′latex−bold′>(A)</span>164<spanclass=′latex−bold′>(B)</span>16<spanclass=′latex−bold′>(C)</span>14<spanclass=′latex−bold′>(D)</span>516<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span>\dfrac1{64}\qquad<span class='latex-bold'>(B) </span>\dfrac16\qquad<span class='latex-bold'>(C) </span>\dfrac14\qquad<span class='latex-bold'>(D) </span>\dfrac5{16}\qquad<span class='latex-bold'>(E) </span>\dfrac12
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