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document.write( "MATHEMATICS\r\n" );
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document.write( "There are 2 indistinguishable M's, \r\n" );
document.write( "2 indistinguishable A's and \r\n" );
document.write( "2 indistinguishable T's.\r\n" );
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document.write( "Case 1: arrangements of 4 distinguishable\r\n" );
document.write( "letters, such as TIME and HEAT.\r\n" );
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document.write( "There are 8 distinguishable letters from\r\n" );
document.write( "this set: {M,A,T,H,E,I,C,S}\r\n" );
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document.write( "Choose the 1st letter 8 ways.\r\n" );
document.write( "Choose the 2nd letter 7 ways.\r\n" );
document.write( "Choose the 3rd letter 6 ways.\r\n" );
document.write( "Choose the 4th letter 5 ways.\r\n" );
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document.write( "That's 8*7*6*5 = 8P4 = 1680 arrangements\r\n" );
document.write( "for case 1.\r\n" );
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document.write( "Case 2: arrangements of exactly one pair of\r\n" );
document.write( "indistinguishable letters, such as MASS or THAT.\r\n" );
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document.write( "Choose the letter for the pair of\r\n" );
document.write( "indistinguishable letters 3 ways,\r\n" );
document.write( "from (M,A,T}\r\n" );
document.write( "Choose the 2 positions in the arrangement\r\n" );
document.write( "from the set of positions:\r\n" );
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document.write( "{1st letter, 2nd letter, 3rd letter, 4th\r\n" );
document.write( "letter} \r\n" );
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document.write( "for the pair of indistinguishable letters to\r\n" );
document.write( "go 4C2 = 6 ways.\r\n" );
document.write( "Choose the letter for the left-most\r\n" );
document.write( "unfilled position 7 ways.\r\n" );
document.write( "Choose the letter for the rightmost-most\r\n" );
document.write( "unfilled position 6 ways.\r\n" );
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document.write( "That's 3*6*7*6 = 756 ways.\r\n" );
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document.write( "Case 3: arrangement of two pair of \r\n" );
document.write( "distinguishable letters, such as MAMA or TATA.\r\n" );
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document.write( "Choose the two letters from {M,A,T} for the\r\n" );
document.write( "pairs of indistinguishable.\r\n" );
document.write( "That's 3C2 = 3 ways.\r\n" );
document.write( "Choose the positions for the pair in 4C2=6\r\n" );
document.write( "ways.\r\n" );
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document.write( "That's 3*6 or 18 ways.\r\n" );
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document.write( "Grand total: 1680+756+18 = 2454 \r\n" );
document.write( "distinguishable 4-letter arrangements from\r\n" );
document.write( "MATHEMATICS\r\n" );
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document.write( "Edwin \n" );
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