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Grogg, Lizzie, Alex, Winnie, and Max each own a hat marked with the first letter of their name. All five hats are placed in a bag. Each beast then removes one hat from the bag in turn at random and places it on their head. Once this is done, what is the probability that exactly two beasts are wearing hats marked with the first letters of their names?

Aug 4, 2023

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There are a total of 5! ways for the beasts to draw the hats, since there are 5 distinct hats and 5 beasts.

There are two ways to get exactly two beasts wearing hats marked with the first letters of their names. Either Grogg and Lizzie both get their hats, or Alex and Winnie both get their hats.

In the case of Grogg and Lizzie, there are 4! ways for the other three beasts to draw hats, since there are 4 hats left and 3 beasts. So there are 4! = 24 ways for Grogg and Lizzie to both get their hats.

Similarly, there are 4! = 24 ways for Alex and Winnie to both get their hats.  So there are a total of 24+24 = 48 ways for exactly two beasts to get their hats.

Therefore, the probability that exactly two beasts get their hats is 48/5! = 2/5.

Aug 4, 2023