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Answer 9!/3!2!=30240 Note there are three a's and two p's and a total of 9 letters in apparatus. So there are 9 choices for the first letter, and 8 for the next, then 7 for the third etc. The we divide by 3! and 2! to avoid overcounting. The answer is 9!/3!2! The numerator can be written as 9! which is pronounced 9 factorial. Then we divide that by 3! times 2! In general if you have have n letters, there are n! words you can make, However, if some of the letters are repeated you must divide to avoid overcounting. If letter a is repeated r1 times, b r2 times, c r3 times.. etc, Then the number of words is n!/r1!r2!r3!

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Q: How many different nine letter arrangements can be formed from the nine letters of the word apparatus?
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