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How many three-letter words can be made from the first eight letters of the alphabet if consonants cannot be next to each other and letters cannot be repeated?
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document.write( "The other tutor's answer is wrong. \r\n" );
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document.write( "Consonants: {B, C, D, F, G, H}\r\n" );
document.write( "Vowels: {A, E} \r\n" );
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document.write( "There are four configurations so that consonants don't come together: \r\n" );
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document.write( "1. Consonant, Vowel, Consonant\r\n" );
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document.write( "2. Consonant, Vowel, Vowel\r\n" );
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document.write( "3. Vowel, Vowel, Consonant\r\n" );
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document.write( "4. Vowel, Consonant, Vowel\r\n" );
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document.write( "For configuration 1, there are 6 ways to choose the first letter as \r\n" );
document.write( "a consonant. For each of those 6 ways to choose the first letter as\r\n" );
document.write( "a consonant there remain 5 ways to choose the third letter as a \r\n" );
document.write( "different consonant. That's 6*5 ways to choose the first and third\r\n" );
document.write( "letters as different consonants. For each of these 6*5 ways to\r\n" );
document.write( "choose the first and third letters as different consonats, there are\r\n" );
document.write( "2 ways to choose the middle letter as a vowel. That's 6*5*2 or 60 ways.\r\n" );
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document.write( "For each of the other three possible configurations, there are two\r\n" );
document.write( "vowels and a consonant. In each case there are 6 ways to choose the one\r\n" );
document.write( "consonant. For each of those 6 ways to choose the consonant there are 2\r\n" );
document.write( "ways to choose the first vowel vowel. That's 6*2 ways to choose the \r\n" );
document.write( "consonant and the leftmost vowel. For each of those 6*2 ways to choose\r\n" );
document.write( "the consonant and the leftmost vowel, there is only 1 way to choose\r\n" );
document.write( "the rightmost vowel. That's 6*2*1 or 12 ways.\r\n" );
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document.write( "Since there are 3 such configurations, and 12 ways to choose each, That's\r\n" );
document.write( "3*12 or 36 ways to choose configurations 2 3 and 4. So the total number of ways is:\r\n" );
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document.write( "60 + 36 or 96 ways.\r\n" );
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document.write( "Here they all are:\r\n" );
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document.write( "1. AEB\r\n" );
document.write( "2. AEC\r\n" );
document.write( "3. AED\r\n" );
document.write( "4. AEF\r\n" );
document.write( "5. AEG\r\n" );
document.write( "6. AEH\r\n" );
document.write( "7. ABE\r\n" );
document.write( "8. ACE\r\n" );
document.write( "9. ADE\r\n" );
document.write( "10. AFE\r\n" );
document.write( "11. AGE\r\n" );
document.write( "12. AHE\r\n" );
document.write( "13. EAB\r\n" );
document.write( "14. EAC\r\n" );
document.write( "15. EAD\r\n" );
document.write( "16. EAF\r\n" );
document.write( "17. EAG\r\n" );
document.write( "18. EAH\r\n" );
document.write( "19. EBA\r\n" );
document.write( "20. ECA\r\n" );
document.write( "21. EDA\r\n" );
document.write( "22. EFA\r\n" );
document.write( "23. EGA\r\n" );
document.write( "24. EHA\r\n" );
document.write( "25. BAE\r\n" );
document.write( "26. BAC\r\n" );
document.write( "27. BAD\r\n" );
document.write( "28. BAF\r\n" );
document.write( "29. BAG\r\n" );
document.write( "30. BAH\r\n" );
document.write( "31. BEA\r\n" );
document.write( "32. BEC\r\n" );
document.write( "33. BED\r\n" );
document.write( "34. BEF\r\n" );
document.write( "35. BEG\r\n" );
document.write( "36. BEH\r\n" );
document.write( "37. CAE\r\n" );
document.write( "38. CAB\r\n" );
document.write( "39. CAD\r\n" );
document.write( "40. CAF\r\n" );
document.write( "41. CAG\r\n" );
document.write( "42. CAH\r\n" );
document.write( "43. CEA\r\n" );
document.write( "44. CEB\r\n" );
document.write( "45. CED\r\n" );
document.write( "46. CEF\r\n" );
document.write( "47. CEG\r\n" );
document.write( "48. CEH\r\n" );
document.write( "49. DAE\r\n" );
document.write( "50. DAB\r\n" );
document.write( "51. DAC\r\n" );
document.write( "52. DAF\r\n" );
document.write( "53. DAG\r\n" );
document.write( "54. DAH\r\n" );
document.write( "55. DEA\r\n" );
document.write( "56. DEB\r\n" );
document.write( "57. DEC\r\n" );
document.write( "58. DEF\r\n" );
document.write( "59. DEG\r\n" );
document.write( "60. DEH\r\n" );
document.write( "61. FAE\r\n" );
document.write( "62. FAB\r\n" );
document.write( "63. FAC\r\n" );
document.write( "64. FAD\r\n" );
document.write( "65. FAG\r\n" );
document.write( "66. FAH\r\n" );
document.write( "67. FEA\r\n" );
document.write( "68. FEB\r\n" );
document.write( "69. FEC\r\n" );
document.write( "70. FED\r\n" );
document.write( "71. FEG\r\n" );
document.write( "72. FEH\r\n" );
document.write( "73. GAE\r\n" );
document.write( "74. GAB\r\n" );
document.write( "75. GAC\r\n" );
document.write( "76. GAD\r\n" );
document.write( "77. GAF\r\n" );
document.write( "78. GAH\r\n" );
document.write( "79. GEA\r\n" );
document.write( "80. GEB\r\n" );
document.write( "81. GEC\r\n" );
document.write( "82. GED\r\n" );
document.write( "83. GEF\r\n" );
document.write( "84. GEH\r\n" );
document.write( "85. HAE\r\n" );
document.write( "86. HAB\r\n" );
document.write( "87. HAC\r\n" );
document.write( "88. HAD\r\n" );
document.write( "89. HAF\r\n" );
document.write( "90. HAG\r\n" );
document.write( "91. HEA\r\n" );
document.write( "92. HEB\r\n" );
document.write( "93. HEC\r\n" );
document.write( "94. HED\r\n" );
document.write( "95. HEF\r\n" );
document.write( "96. HEG\r\n" );
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document.write( "But only these 18: \r\n" );
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document.write( "ACE, ADE, AGE, BAD, BAG, BAH, BED, BEG, CAB,\r\n" );
document.write( "CAD, DAB, FAD, FAG, FED, GAB, GAD, HAD, and HAG \r\n" );
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document.write( "are REAL words. :-)\r\n" );
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document.write( "Edwin \n" );
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