A bag contains 6 Red and 6 black balls. 6 pair of balls are selected one by one without replacement then the probability that each of the 6 pairs contains 1 red and 1 black ball.
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For problems involving "each pair contains one of each type," think of it as a matching problem. The number of ways to match $n$ items of type A with $n$ items of type B is always $n!$.