How many ways can I arrange the letters of the word BANANA?

You are right that the repeated letters are the key. The word BANANA has 6 letters, but the letter A appears 3 times and N appears 2 times, while B appears once. If all letters were different the answer would be 6 factorial = 720. But swapping identical letters does not create a new arrangement, so we must remove those duplicate counts by dividing by the factorials of the repeats. So the number of distinct arrangements is 6! divided by (3! times 2!), which is 720 divided by (6 times 2) = 720/12 = 60. The general rule for arranging n items where some repeat is n! divided by the product of the factorials of each repeated letter's count. Always check for repeated letters before using plain n!.