Question 1188357

There are {{{7}}} letters, so there are {{{7!}}} ways in arranging {{{7}}}} objects into {{{7}}} spaces. 
But we've overcounted by a factor of {{{3!}}} ({{{a}}}'s can interchange in {{{3!}}} ways), and 
{{{2!}}} ({{{t}}}'s can interchange in {{{2! }}}ways).

To better understand this concept, let's name each repeated letter:
{{{a[1]}}} {{{ t[1]}}} {{{ l}}}  {{{a[2]}}}  {{{n}}} {{{ t[2]}}}{{{  a[3]}}}

Now, we can rearrange Atlanta to make:
{{{a[2]}}} {{{ t[2]}}}  {{{l}}} {{{a[3]}}}  {{{n}}}  {{{t[1]}}}  {{{a[1]}}}

Without the superscripts, we make Atlanta. 

But, we've already made that word with:
{{{a[1]}}} {{{ t[1]}}} {{{ l}}}  {{{a[2]}}}  {{{n}}} {{{ t[2]}}}{{{  a[3]}}}

This is because we need to account for the times the{{{ a}}}'s can change and we wouldn't notice, and the {{{t}}}'s can change and we won't notice.

Thus, our final number of ways becomes:

  {{{7!/(3!2!)=420}}} different ways