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find the no.of combinations of the word INDEPENDENT taking 5 letters at a time.
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document.write( "Wow! What a difficult problem!\r\n" );
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document.write( "There are 2 D's, 3 E's, 1 I, 3 N's, 1 P, and 1 T\r\n" );
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document.write( "I hope you really meant combinations, and not permutations. For I am not\r\n" );
document.write( "considering taking the letters in any order. For example, \r\n" );
document.write( "\"E,E,N,N,N\", \"E,N,E,N,N\", \"N,N,N,E,E\", and \"E,N,N,N,E\" \r\n" );
document.write( "are all counted as the same combination, even though they would be different\r\n" );
document.write( "permutations. Let me know in the thank-you note if you meant permutations\r\n" );
document.write( "instead, and I'll re-work the problem.\r\n" );
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document.write( "There are 5 basic cases of choosing 5 letters. I will use the last five\r\n" );
document.write( "letters of the alphabet, V,W,X,Y and Z as \"placeholders\" for the letters of a\r\n" );
document.write( "combination. Here are the 5 types of combinations:\r\n" );
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document.write( "1. V,V,W,W,W 1 pair, both of the same letters, and 1 triplet, all of the same letter. \r\n" );
document.write( "2. V,V,W,W,X 2 pairs of same letters and 1 non-matching letter.\r\n" );
document.write( "3. V,V,W,X,Y 1 pair of the same letter and 3 non-matching letters. \r\n" );
document.write( "4. V,V,V,X,Y 1 triplet, all of the same letter and 2 non-matching letters.\r\n" );
document.write( "5. V,W,X,Y,Z 5 non-matching letters.\r\n" );
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document.write( "Case 1. V,V,W,W,W\r\n" );
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document.write( "W is more restrictive here than V so we choose it first.\r\n" );
document.write( "We can choose the letter for W in 2 ways, either E or N\r\n" );
document.write( "We can choose the letter for V in 2 ways, D or whichever one of {E,N} we didn't\r\n" );
document.write( "choose in the preceding sentence.\r\n" );
document.write( "That's 2x2 or 4 combinations for case 1.\r\n" );
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document.write( "Case 2. V,V,W,W,X\r\n" );
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document.write( "We can choose the 2 letters for the 2 pairs from {D,E,N} in C(3,2) = 3 ways\r\n" );
document.write( "We can choose the letter for X in any of the remaining 4 ways.\r\n" );
document.write( "That's 3x4 or 12 combinations for case 2.\r\n" );
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document.write( "Case 3. V,V,W,X,Y\r\n" );
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document.write( "We can choose the letter for the pair 3 ways, D,E, or N\r\n" );
document.write( "We can choose the letters W,X,Y from any of the 5 remaining ways, in C(5,3) or\r\n" );
document.write( "10 ways\r\n" );
document.write( "That 3x10 or 30 combinations for case 3.\r\n" );
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document.write( "Case 4. V,V,V,X,Y\r\n" );
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document.write( "We can choose the letter for V in 2 ways, E or N.\r\n" );
document.write( "We can choose X,Y from any of the remaining 5 letters, in C(5,3) or 10 ways.\r\n" );
document.write( "That's 2x10 or 20 combinations for case 4.\r\n" );
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document.write( "Case 5. V,W,X,Y,Z\r\n" );
document.write( "That's 6 letters taken 5 at a time, or C(6,5) = 6 ways\r\n" );
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document.write( "Answer: 4+12+30+20+6 = 72\r\n" );
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document.write( "Edwin \n" );
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