I have letters AAABCD. The letters A and B must be together. How many ways can I arrange these 6 letters?
At first, I thought the answer would be to group one of the A's and B together, and treat it as one object, so that's 5! ways of arranging {A,B} A A C D. Then divided by 2! to account for the repeated A's outside of the group then multiplied by 2! to account for the number of ways one can arrange the A and B inside the group.
However, I later realized that certain permutations are in fact duplicates, e.g. {A,B}AACD and A{B,A}ACD and the number of duplicates seem to be n! where n is the number of repeated objects, however I'm not sure how to prove this.
Best Answer
The duplicates can be excluded by counting the permutations of four symbols : $ABA$, $A$, $C$, $D$ which is simply $4!$
Desired answer is $5!-4!=96$