document.write( "Question 1188357: In how many distinct ways can the letters of the world ATLANTA be arranged?
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Algebra.Com's Answer #819426 by MathLover1(20849) $\"\"$ $\"About$

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\n" ); document.write( "There are $\"7\"$ letters, so there are $\"7%21\"$ ways in arranging $\"7\"$ } objects into $\"7\"$ spaces.
\n" ); document.write( "But we've overcounted by a factor of $\"3%21\"$ ( $\"a\"$ 's can interchange in $\"3%21\"$ ways), and
\n" ); document.write( " $\"2%21\"$ ( $\"t\"$ 's can interchange in $\"2%21+\"$ ways).\r
\n" ); document.write( "\n" ); document.write( "To better understand this concept, let's name each repeated letter:
\n" ); document.write( " $\"a%5B1%5D\"$ $\"+t%5B1%5D\"$ $\"+l\"$ $\"a%5B2%5D\"$ $\"n\"$ $\"+t%5B2%5D\"$ $\"++a%5B3%5D\"$ \r
\n" ); document.write( "\n" ); document.write( "Now, we can rearrange Atlanta to make:
\n" ); document.write( " $\"a%5B2%5D\"$ $\"+t%5B2%5D\"$ $\"l\"$ $\"a%5B3%5D\"$ $\"n\"$ $\"t%5B1%5D\"$ $\"a%5B1%5D\"$ \r
\n" ); document.write( "\n" ); document.write( "Without the superscripts, we make Atlanta. \r
\n" ); document.write( "\n" ); document.write( "But, we've already made that word with:
\n" ); document.write( " $\"a%5B1%5D\"$ $\"+t%5B1%5D\"$ $\"+l\"$ $\"a%5B2%5D\"$ $\"n\"$ $\"+t%5B2%5D\"$ $\"++a%5B3%5D\"$ \r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "Thus, our final number of ways becomes:\r
\n" ); document.write( "\n" ); document.write( " $\"7%21%2F%283%212%21%29=420\"$ different ways \r
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