Question

Find numbers distinguishable permutations of the letters ALABAMA_______

Answer 1

n = 7
a is repeated 4 times
so
= n!/r!
= 7!/4!
= 7×6×5×4!/4!
= 7×6×5
= 210

Answer 2

If "ALABAMA" were written so that the A's were of different colors, the number of permutations would be 7!. However, since the four A's look exactly alike in "ALABAMA", the number of distinguishable permutations is much smaller. So what we do is start with the 7! arrangements of "ALABAMA", and divide by the number of ways the four A's can be arranged within each permutation, so that in effect they will all be counted only once.
So the answer is
=7!/4!
=5040/24
= 210.