MathDB
Can I put the balls in the bins

Source: AMC 10A #21/12A #18

November 11, 2021

Problem Statement

Each of the 2020 balls is tossed independently and at random into one of the 55 bins. Let pp be the probability that some bin ends up with 33 balls, another with 55 balls, and the other three with 44 balls each. Let qq be the probability that every bin ends up with 44 balls. What is pq\frac{p}{q}?
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 4<spanclass=latexbold>(C)</span> 8<spanclass=latexbold>(D)</span> 12<spanclass=latexbold>(E)</span> 16<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 16