42.No Cup on Its Own Saucer
★★★combinatoricsJane Street
Six saucers sit on a table: two red, two white and two patterned with stars. Six cups — likewise two red, two white and two starred — are placed on them at random, one cup per saucer, with every placement equally likely. What is the probability that no cup ends up on a saucer of its own pattern?
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Rather than tracking individual cups, count how many cups of each colour land on saucers of each colour. That is a small table of numbers whose row and column sums are forced.
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