Question

In how many distinct ways can the letters of the word ATLANTA be arranged?

Answer 1

7!/3!2!

Answer 2

how did you get that

Answer 3

If all the letters were unique letters, you could arrange it in 7! possible ways. Because we have 3 A's and 2 T's, we then account for those repeating cases by dividing off the ways to combine those 3 A's and 2 T's by themselves. So, 3! ways to arrange the 3 A's with each other, and 2! to arrange the 2 T's.

Answer 4

Repeating cases being such as:
\( \color{red}ATL\color{green}ANT\color{blue}A\)
\( \color{green}ATL\color{red}ANT\color{blue}A\)
\( \color{blue}ATL\color{red}ANT\color{Green}A\)
and 3 other possibilities arranging those A's. This works for any possible arrangement so we divide off the 3! from 7!. It is similar for the T's as well.

Answer 5

so you divided

Answer 6

Yup! Removing repeat cases is always through division, because there are 3! arrangements of A that are the same per combination of the 7 letters and 2! arrangements of T. 7! / (2! * 3!) (both 2! and 3! divided from 7!)