document.write( "Question 911470: how many words can be formed using letter of 'STUDENT' using each letter at most once if some or all the letters may be omitted \n" ); document.write( "
Algebra.Com's Answer #553113 by AnlytcPhil(1806)\"\" \"About 
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I will assume the two T's are indistinguishable and \r\n" );
document.write( "that there can be two T's in the same word.  \r\n" );
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document.write( "The number of words that have all different letters is\r\n" );
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document.write( "6P1 + 6P2 + 6P3 + 6P4 + 6P5 + 6P6 = \r\n" );
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document.write( "6 + 30 + 120 + 360 + 720 + 720 = 1956\r\n" );
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document.write( "In addition to that number, we must calculate:\r\n" );
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document.write( "A. The number of 2 letter words with 2 T's\r\n" );
document.write( "There's only 1, which is TT.\r\n" );
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document.write( "B. The number of 3 letter distinguishable words with 2 T's:\r\n" );
document.write( "There are 3C2 = 3 positions to place the 2 T's. \r\n" );
document.write( "There are 5P1 = 5 ways to place the 1 non-T.\r\n" );
document.write( "That's 3*5 = 15 ways. \r\n" );
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document.write( "C. The number of 4 letter distinguishable words with 2 T's:\r\n" );
document.write( "There are 4C2 = 6 positions to place the 2 T's. \r\n" );
document.write( "There are 5P2 = 20 ways to place the 2 non-T's.\r\n" );
document.write( "That's 6*20 = 120 ways.\r\n" );
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document.write( "D. The number of 5 letter distinguishable words with 2 T's:\r\n" );
document.write( "There are 5C2 = 10 positions to place the 2 T's. \r\n" );
document.write( "There are 5P3 = 60 ways to place the 3 non-T's.\r\n" );
document.write( "That's 10*60 = 600 ways.\r\n" );
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document.write( "E. The number of 6 letter distinguishable words with 2 T's:\r\n" );
document.write( "There are 6C2 = 15 positions to place the 2 T's. \r\n" );
document.write( "There are 5P4 = 120 ways to place the 4 non-T's.\r\n" );
document.write( "That's 15*120 = 1800 ways.\r\n" );
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document.write( "F. The number of 7 letter distinguishable words with 2 T's:\r\n" );
document.write( "There are 7C2 = 21 positions to place the 2 T's. \r\n" );
document.write( "There are 5P5 = 120 ways to place the 5 non-T's.\r\n" );
document.write( "That's 21*120 = 2520 ways.\r\n" );
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document.write( "[As a check on that last number of ways, we can use the usual \r\n" );
document.write( "formula for the number of distinguishable words that can be \r\n" );
document.write( "formed from a 7-letter word with 2 indistinguishable letters, \r\n" );
document.write( "7!/2! = 5040/2 = 2520]\r\n" );
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document.write( "The total number of words with two T's is\r\n" );
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document.write( "1 + 15 + 120 + 600 + 1800 + 2520 = 5056\r\n" );
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document.write( "Grand total = 1956 + 5056 = 7012 \r\n" );
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document.write( "Edwin
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