document.write( "Question 1203273: The letters of the word PROBABILITY are arranged at random. Find the number of arrangements where the two Is are separated. \n" ); document.write( "
Algebra.Com's Answer #838712 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "The letters of the word PROBABILITY are arranged at random.
\n" ); document.write( "Find the number of arrangements where the two Is are separated.
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document.write( "The word PROBABILITY consists of 11 symbols.\r\n" );
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document.write( "Of them, two letters \"B\" make one repeating pair.\r\n" );
document.write( "         Two letters \"I\" make another repeating pair.\r\n" );
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document.write( "The total number of all distinguishable arrangements of the word PROBABILITY is \"11%2F%282%21%2A2%21%29\" = 9,979,200.\r\n" );
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document.write( "Now let's calculate the number of all distinguishable arrangements of the word PROBABILITY\r\n" );
document.write( "where two Is are together (are glued).\r\n" );
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document.write( "This pair of Is, placed together (glued), we can consider as one object.\r\n" );
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document.write( "Then we have arrangements of 10 = 11-1 objects with one repeating pair of Bs.\r\n" );
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document.write( "So, the number of all such distinguishable arrangements is \"10%21%2F2%21\" = 1,814,400. \r\n" );
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document.write( "Finally, the number of all distinguishable arrangements of the word PROBABILITY, where \r\n" );
document.write( "the two Is are separated is the difference\r\n" );
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document.write( "    \"11%21%2F%282%21%2A2%21%29\" - \"10%21%2F2%21\" = 9,979,200 - 1,814,400 = 8,164,800.\r\n" );
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document.write( "ANSWER.  The number of all distinguishable arrangements of the word PROBABILITY, where \r\n" );
document.write( "         the two Is are separated is  \"11%21%2F%282%21%2A2%21%29\" - \"10%21%2F2%21\" = 8,164,800.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The solution by tutor @greenestamps is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "In his solution, he correctly determined the number of all distinguishable arrangements
\n" ); document.write( "of the word PROBABILITY, but mistakenly \"highlight%28highlight%28doubled%29%29\" the number of all distinguishable arrangements
\n" ); document.write( "of this word with two glued Is.\r
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\n" ); document.write( "\n" ); document.write( "MEMORIZE: distinguishable arrangements are not the same as permutations !\r
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\n" ); document.write( "\n" ); document.write( "Do not mix these two different conceptions - - - always DISTINCT them !\r
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\n" ); document.write( "\n" ); document.write( "If you want to see many other similar  (and different)  solved problems of this type,  look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Arranging elements of sets containing indistinguishable elements \r
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\n" ); document.write( "\n" ); document.write( "Learn the subject from there.\r
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