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document.write( "In the word \"examination\",\r\n" );
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document.write( "there are two each of letters 'a', 'i', and 'n', which are indistinguishable,\r\n" );
document.write( "and there is one each of letters 'e', 'o', 'm', 't', and 'x'.\r\n" );
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document.write( "Case 1. Those permutations which contain a maximum of 1 letter from\r\n" );
document.write( "{a,e,i,m,n,o,t,x} which is P(8,5). That's P(8,5) permutations.\r\n" );
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document.write( "The following three cases 2,3,and 4 yield the same number of permutations \r\n" );
document.write( "each:\r\n" );
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document.write( "Case 2. Those which contain two a's, and a maximum of one 'i' or 'n'\r\n" );
document.write( "We choose the positions for the two a's C(5,2) ways and the remaining 3\r\n" );
document.write( "from {e,i,m,n,o,t,x} in P(7,3) ways. That's C(5,2)P(7,3) permutations.\r\n" );
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document.write( "Case 3. Those which contain two i's, and a maximum of one 'a' or 'n'\r\n" );
document.write( "We choose the positions for the two i's C(5,2) ways and the remaining 3\r\n" );
document.write( "from {a,e,m,n,o,t,x} in P(7,3) ways. That's C(5,2)P(7,3) permutations.\r\n" );
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document.write( "Case 4. Those which contain two n's, and a maximum of one 'a' or 'i'\r\n" );
document.write( "We choose the positions for the two n's C(5,2) ways and the remaining 3\r\n" );
document.write( "from {a,e,i,m,o,t,x} in P(7,3) ways. That's C(5,2)P(7,3) permutations.\r\n" );
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document.write( "The following three cases 5,6, and 7 yield the same number of permutations\r\n" );
document.write( "each:\r\n" );
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document.write( "Case 5. Those which contain two a's and two i's, and a maximum of one 'n'\r\n" );
document.write( "We choose the positions for the two a's C(5,2), and the positions for the\r\n" );
document.write( "two i's C(3,2), and one letter from {e,m,n,o,t,x} in P(6,1) ways. That's\r\n" );
document.write( "C(5,2)C(3,2)P(6,1) ways.\r\n" );
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document.write( "Case 6. Those which contain two a's and two n's, but a maximum of one i\r\n" );
document.write( "We choose the positions for the two a's C(5,2), and the positions for the\r\n" );
document.write( "two n's C(3,2), and one letter from {e,i,m,o,t,x} in P(6,1) ways. That's\r\n" );
document.write( "C(5,2)C(3,2)P(6,1) ways.\r\n" );
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document.write( "Case 7. Those which contain two i's and two n's, but a maximum of one 'a'\r\n" );
document.write( "We choose the positions for the two i's C(5,2), and the positions for the\r\n" );
document.write( "two n's C(3,2), and one letter from {a,e,m,o,t,x} in P(6,1) ways. That's\r\n" );
document.write( "C(5,2)C(3,2)P(6,1) ways.\r\n" );
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document.write( "Total = P(8,5) + 3·C(5,2)P(7,3) + 3·C(5,2)C(3,2)P(6,1) = \r\n" );
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document.write( "6720 + 3·10·210 + 3·10·3·6 = 6720 + 6300 + 540 = 13560.\r\n" );
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document.write( "Answer: 13,560\r\n" );
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document.write( "Edwin\n" ); document.write( " \n" ); document.write( "