In how many different ways can the letters in the letter in the word ARRANGEMENT be arranged?

If the letters were all different the answer would be 11! but since the word contains some duplicates you need to reduce this. Here's how - note that there are 2 A's, so for each arrangement of the A's there are two "duplicates," hence you divide by 2. In general for each letter that occurs N times you divide by N! because that's how many ways those duplicates could be arranged. For example the word TEETHE has 2 T's and 3 E's, so the number of unique arrangements is 6!/(2! X 3!) = 60.