How many distinguishable ways can you arrange the letters aaabb?

Answer 1

To find the number of distinguishable arrangements of the letters "aaabb," we use the formula for permutations of multiset:

[ \frac{n!}{n_1! \times n_2!} ]

where ( n ) is the total number of letters, ( n_1 ) is the number of indistinguishable letters of one type, and ( n_2 ) is the number of indistinguishable letters of another type. Here, ( n = 5 ) (total letters), ( n_1 = 3 ) (for 'a'), and ( n_2 = 2 ) (for 'b').

Calculating this gives:

[ \frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10 ]

Thus, there are 10 distinguishable ways to arrange the letters "aaabb."