How do you find the number of distinguishable permutations of the group of letters: A, A, G, E, E, E, M?
#A, A, G, E, E, E, M#
For the sake of discussion, let's distinguish all of the letters by adding subscripts:
#A_1, A_2, G, E_1, E_2, E_3, M#
The number of distinguishable permutations of these marked letters is:
#7! = 7*6*5*4*3*2*1 = 5040#
If we now remove the marking, then some of these distinguishable permutations become indistinguishable.
In fact, each of the previously distinguishable arrangements has a total of
So the total number of distinguishable permutations is:
#(7!)/(2!3!) = 5040/(2*6) = 420#