How many ordered lists of length 10 using letters from {A, B, C} use exactly 5 "A"s?

You have 10 letters and 5 have to be "A"s.





Use combinatorics...





10 choose 5 = 252





There are 252 different ways of filling A's in 5 slots.





Now you need to find how many different ways of filling B's and C's in the other five, which is simply:





2 x 2 x 2 x 2 x 2 = 32





252 x 32 = 8064





There are 8064.

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