document.write( "Question 1147433: How many more possible arrangements can be made using the letters of the word PERMUTATIONS than using the letters of the word COMBINATIONS?  \n" ); document.write( "
Algebra.Com's Answer #768758 by ikleyn(52781)\"\" \"About 
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document.write( "Word \"PERMUTATIONS\" has 12 symbols, of them \"T\" has multiplicity 2, and the rest symbols have multiplicity 1.\r\n" );
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document.write( "Therefore, the number of distinguishable arrangements of the word \"PERMUTATIONS\" is\r\n" );
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document.write( "    \"N%5B1%5D\" = \"12%21%2F2%21\" = \"%2812%2A11%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%282%2A1%29\" = 239500800.     (1)\r\n" );
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document.write( "In the formula (1), 2! in the denominator serves to account for repeating \"T\" in the word.\r\n" );
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document.write( "Word \"COMBINATIONS\" has 12 symbols, of them \r\n" );
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document.write( "    - the letter \"O\" has multiplicity 2,\r\n" );
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document.write( "    - the letter \"I\" has multiplicity 2,\r\n" );
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document.write( "    - and all other letters are not repeating.\r\n" );
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document.write( "Therefore, the number of distinguishable arrangements of the word \"COMBINATIONS\" is\r\n" );
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document.write( "    \"N%5B2%5D\" = \"12%21%2F%282%21%2A2%21%29\" = \"%2812%2A11%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%281%2A2%2A1%2A2%29\" = 119750400.     (2)\r\n" );
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document.write( "In the formula (2), two factors  of  2!  in the denominator serves to account for repeating letters \"O\"  and  \"I\"  in the word.\r\n" );
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document.write( "The difference  \"N%5B1%5D\" - \"N%5B2%5D\" = 239500800 - 119750400 = 119750400.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "On Permutations,  see the lessons \r
\n" ); document.write( "\n" ); document.write( "    - Introduction to Permutations\r
\n" ); document.write( "\n" ); document.write( "    - PROOF of the formula on the number of Permutations\r
\n" ); document.write( "\n" ); document.write( "    - Problems on Permutations\r
\n" ); document.write( "\n" ); document.write( "    - Arranging elements of sets containing indistinguishable elements \r
\n" ); document.write( "\n" ); document.write( "    - Persons sitting around a cicular table \r
\n" ); document.write( "\n" ); document.write( "    - Combinatoric problems for entities other than permutations and combinations \r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on permutations, combinations and other combinatoric entities \r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on Permutations and Combinations\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Combinatorics: Combinations and permutations\". \r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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