Final answer:
The number of distinguishable permutations of the letters A A B C D is 60, as determined by the formula for permutations with identical objects 5!/(2!)
Step-by-step explanation:
To find the number of distinguishable permutations of the given letters A A B C D, we use the formula for permutations of a set of objects where some objects are identical. The general formula for the number of distinguishable permutations of n objects, where there are n1 objects of one type, n2 objects of another type, and so on, is:
n!/(n1! * n2! * ... * nk!).
In this case, we have 5 objects (letters) in total with 2 of them being identical (the 'A's). Therefore, the number of distinguishable permutations is 5!/(2!), which is (5 * 4 * 3 * 2 * 1)/(2 * 1). This simplifies to 60.
So, there are 60 distinguishable permutations for the letters A A B C D.