2022 AIME I Problems/Problem 9
Ellina has twelve blocks, two each of red (), blue (), yellow (), green (), orange (), and purple (). Call an arrangement of blocks if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is where and are relatively prime positive integers. Find
Consider this position chart: Since there has to be an even number of spaces between each pair of the same color, spots , , , , , and contain some permutation of all colored balls. Likewise, so do the even spots, so the number of even configurations is (after putting every pair of colored balls in opposite parity positions, the configuration can be shown to be even). This is out of possible arrangements, so the probability is: which is in simplest form. So, .
We can simply use constructive counting. First, let us place the red balls; choose the first slot in ways, and the second in ways, because the number is cut in half due to the condition in the problem. This gives ways to place the red balls. Similarly, there are ways to place the blue balls, and so on, until there are ways to place the purple balls. Thus, the probability is and the desired answer extraction is .
Use constructive counting, as per above. WLOG, place the red blocks first. There are 11 ways to place them with distance 0, 9 ways them to place with distance 2, so on, so the way to place red blocks is . Then place any other block similarly, with ways (basic counting). You get then ways to place the blocks evenly, and ways to place the blocks in any way, so you get by simplifying.
Video Solution (Mathematical Dexterity)
Video Solution (Power of Logic)
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