How many unique ways are there to arrange the letters in the word HATTER?
I can't wrap my head around the math to find the answer. I know that if they were all different letters the answer would be 6!. However, I know that these T's are going to overlap, so it won't be that.
I am trying to give myself examples like AAA, it can only be written once but if it was 3 different letters it would be 6 times instead. Somehow I need to get a 6/6, so that it can become 1.
If I try it with AAC, half of the permutations disappear. So it must be divided by 2 I guess. 6/2.
- ABC AAC 1
- ACB ACA 2
- BCA ACA 2
- BAC AAC 1
- CAB CAA 3
- CBA CAA 3
I kind of see a pattern here. Possible combinations if all letters were different factorial / Divide by the number of equal letters factorial, but still I am confused.
Explanation is appreciated.
The answer is 360.
Best Answer
Imagine one of the Ts is red and the other is blue. Then write out all 6!=720 arrangements. You will see that while they are all unique, you can create pairs where the only difference is the position of the red and blue Ts. Since they are identical in the original question, you must divide by the number of ways the Ts can be arranged. In this case, $2!=2$. So your answer is $$\frac{6!}{2!}$$
Generally you can write the answer as the total number of letters factorial divided by the count of each letter factorial. In this case $$\frac{6!}{2!\cdot 1!\cdot1!\cdot1!\cdot1!}$$