SOLUTION: In how many distinct ways can the letters of the world ATLANTA be arranged?

Algebra.Com
Question 1188357: In how many distinct ways can the letters of the world ATLANTA be arranged?
Found 2 solutions by Edwin McCravy, MathLover1:
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
If all 7 letters were distinguishable, the answer would be 7!
But since the 3 A's are not distinguishable, 3! of the 7! look alike, so we divide by 3!, and get 7!/3!

Since the 2 T's are not distinguishable, 2! of the 7!/3! look alike, so we also divide by 2!

Answer: 

Edwin
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

There are letters, so there are ways in arranging } objects into spaces.
But we've overcounted by a factor of ('s can interchange in ways), and
('s can interchange in ways).
To better understand this concept, let's name each repeated letter:

Now, we can rearrange Atlanta to make:

Without the superscripts, we make Atlanta.
But, we've already made that word with:

This is because we need to account for the times the's can change and we wouldn't notice, and the 's can change and we won't notice.
Thus, our final number of ways becomes:
different ways

RELATED QUESTIONS

In how many distinct ways can the letters of the word ROBBER be... (answered by Timnewman)
In how many distinct ways can the letters of the word SEEDS be... (answered by fractalier)
How many distinct ways can the letters in the word ITEMS be arranged? How many distinct (answered by addingup)
In how many distinct ways can the letters of the word LLAMAS be​... (answered by ewatrrr)
Using all ten letters of the word “thermastar”, how many DISTINCT ways can these letters... (answered by sudhanshu_kmr)
In how many distinct ways can the TORONTO be... (answered by ewatrrr,Theo)
How many distinct ways can the letters in the word ITEMS be arranged? How many distinct (answered by ikleyn,Alan3354)
how many ways can the letters in 'Alaska' be... (answered by ad_alta)
In how many ways can the letters QUIZ be... (answered by ewatrrr)