document.write( "Question 175393: How many different permutations can you make with the letters in the word seventeen? \n" ); document.write( "

Algebra.Com's Answer #130535 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Warning: Gonzo's solution is incorrect!
\n" ); document.write( "Edwin's solution:\r
\n" ); document.write( "\n" ); document.write( "How many different permutations can you make with the letters in the word seventeen
\n" ); document.write( "

\r\n" );
document.write( "There are two methods for finding this:\r\n" );
document.write( "\r\n" );
document.write( "Method 1:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and \r\n" );
document.write( "\r\n" );
document.write( "Method 2:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Explanation of method 1:\r\n" );
document.write( "\r\n" );
document.write( "Since \"seventeen\" is a 9-letter word, there are 9 positions \r\n" );
document.write( "to place the letters.\r\n" );
document.write( "\r\n" );
document.write( "We can choose any 1 of the 9 positions for the \"s\". \r\n" );
document.write( "That's 9C1 ways to place the s. After placing it, that\r\n" );
document.write( "leaves 8 positions to fill.\r\n" );
document.write( "\r\n" );
document.write( "We can then choose any 4 of the 8 remaining positions \r\n" );
document.write( "for the 4 \"e\"'s. That's 8C4 ways to place the 4 e's. After \r\n" );
document.write( "placing them, that leaves 4 positions to fill.\r\n" );
document.write( "\r\n" );
document.write( "We can then choose any 1 of the 4 remaining positions \r\n" );
document.write( "for the \"v\". That's 4C1 ways to place the \"v\". After \r\n" );
document.write( "placing it, that leaves 3 positions to fill.\r\n" );
document.write( "\r\n" );
document.write( "We can then choose any 2 of the 3 remaining positions \r\n" );
document.write( "for the 2 \"n\"'s. That's 3C2 ways to place the 2 \"n\"'s. After \r\n" );
document.write( "placing them, that leaves only 1 position to fill.\r\n" );
document.write( "\r\n" );
document.write( "So we can chose the position for the \"t\" in only 1C1 or 1 way.\r\n" );
document.write( "\r\n" );
document.write( "So the answer using method 1, is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Explanation of Method 2:\r\n" );
document.write( "\r\n" );
document.write( "First pretend that the 4 e's are all different colors, so that we \r\n" );
document.write( "could tell them apart. And also pretend that the 2 n's are different \r\n" );
document.write( "colors, so that we can tell them apart too.\r\n" );
document.write( "\r\n" );
document.write( "That is let's pretend that the word \"seventeen\" is spelled with\r\n" );
document.write( "colored letters, like this\r\n" );
document.write( "\r\n" );
document.write( "seventeen\r\n" );
document.write( "\r\n" );
document.write( "So there would be 9! or 362880 ways to arrange them if we could \r\n" );
document.write( "tell the e's apart and also tell the n's apart.\r\n" );
document.write( "\r\n" );
document.write( "So one random sample of such an arrangment of \"seventeen\" is, \r\n" );
document.write( "say, this:\r\n" );
document.write( "\r\n" );
document.write( "nevtesene\r\n" );
document.write( "\r\n" );
document.write( "Now within this one particular sample arrangement there are 4! ways\r\n" );
document.write( "we can arrange the colors of the e's and there are 2! ways we can\r\n" );
document.write( "arrange the colors of the n's. \r\n" );
document.write( "\r\n" );
document.write( "So that's 4!2! or 48 ways to move the e's and n's around in that \r\n" );
document.write( "one particular sample. And every one of these would be a\r\n" );
document.write( "case of the arrangement \"nevtesene\". \r\n" );
document.write( "\r\n" );
document.write( "Every sample would be the same, so that means that the 9! or 362880\r\n" );
document.write( "counts every arrangement 4!2! or 48 times too many, so we must \r\n" );
document.write( "divide it by 48 to get the correct answer. So the answer, using \r\n" );
document.write( "method 2 is\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );