AMC 10 Daily Practice - Counting
Complete problem set with solutions and individual problem pages
Ang, Ben, and Jasmin each have blocks, colored red,blue, yellow, white, and green; andthere are empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives blocks all of the same color is ,where and are relatively prime positive integers. What is ? (2021 AMC Spring 10B Problems, Question #22)
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Let our denominator be , so we consider all possible distributions.
We use PIE (Principle of Inclusion and Exclusion) to count the successful ones.
When we have at box with all balls the same color in that box, there are ways for the distributions to occur ( for selecting one of the five boxes for a uniform color, for choosing the color for that box, for each of the three people to place their remaining items).
However, we overcounted those distributions where two boxes had uniform color, and there are ways for the distributions to occur ( for selecting two of the five boxes for a uniform color, for choosing the color for those boxes, for each of the three people to place their remaining items).
Again, we need to re-add back in the distributions with three boxes of uniform color\cdots and so on so forth.
Our success by PIE is
yielding an answer of .
