SOLUTION: 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?

Algebra ->  Permutations -> SOLUTION: 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?      Log On


   

Question 371655: 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?
Found 2 solutions by amoresroy, Edwin McCravy:
Answer by amoresroy(361) About Me  (Show Source):
You can put this solution on YOUR website!
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?
The number of three-letter words with consonant as the first letter. The choices are b,c,d,f,g,h.
= 6*2*5 = 60
The number of three-letter words with vowel as the first letter. The choices are a & e
= 2*6*1 = 12
The number of three-letter words that can be made from the first eight letters of the alphabet is 72. (60+12)



Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
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?

The other tutor's answer is wrong. 

Consonants: {B, C, D, F, G, H}
Vowels:     {A, E} 

There are four configurations so that consonants don't come together: 

1.  Consonant, Vowel, Consonant

2.  Consonant, Vowel, Vowel

3.  Vowel, Vowel, Consonant

4.  Vowel, Consonant, Vowel

For configuration 1, there are 6 ways to choose the first letter as 
a consonant. For each of those 6 ways to choose the first letter as
a consonant there remain 5 ways to choose the third letter as a 
different consonant.  That's 6*5 ways to choose the first and third
letters as different consonants.  For each of these 6*5 ways to
choose the first and third letters as different consonats, there are
2 ways to choose the middle letter as a vowel.  That's 6*5*2 or 60 ways.

For each of the other three possible configurations, there are two
vowels and a consonant. In each case there are 6 ways to choose the one
consonant. For each of those 6 ways to choose the consonant there are 2
ways to choose the first vowel vowel.  That's 6*2 ways to choose the 
consonant and the leftmost vowel.  For each of those 6*2 ways to choose
the consonant and the leftmost vowel,  there is only 1 way to choose
the rightmost vowel.  That's 6*2*1 or 12 ways.

Since there are 3 such configurations, and 12 ways to choose each, That's
3*12 or 36 ways to choose configurations 2 3 and 4.  So the total number of ways is:

60 + 36 or 96 ways.

Here they all are:

1. AEB
2. AEC
3. AED
4. AEF
5. AEG
6. AEH
7. ABE
8. ACE
9. ADE
10. AFE
11. AGE
12. AHE
13. EAB
14. EAC
15. EAD
16. EAF
17. EAG
18. EAH
19. EBA
20. ECA
21. EDA
22. EFA
23. EGA
24. EHA
25. BAE
26. BAC
27. BAD
28. BAF
29. BAG
30. BAH
31. BEA
32. BEC
33. BED
34. BEF
35. BEG
36. BEH
37. CAE
38. CAB
39. CAD
40. CAF
41. CAG
42. CAH
43. CEA
44. CEB
45. CED
46. CEF
47. CEG
48. CEH
49. DAE
50. DAB
51. DAC
52. DAF
53. DAG
54. DAH
55. DEA
56. DEB
57. DEC
58. DEF
59. DEG
60. DEH
61. FAE
62. FAB
63. FAC
64. FAD
65. FAG
66. FAH
67. FEA
68. FEB
69. FEC
70. FED
71. FEG
72. FEH
73. GAE
74. GAB
75. GAC
76. GAD
77. GAF
78. GAH
79. GEA
80. GEB
81. GEC
82. GED
83. GEF
84. GEH
85. HAE
86. HAB
87. HAC
88. HAD
89. HAF
90. HAG
91. HEA
92. HEB
93. HEC
94. HED
95. HEF
96. HEG

But only these 18:  

ACE, ADE, AGE, BAD, BAG, BAH, BED, BEG, CAB,
CAD, DAB, FAD, FAG, FED, GAB, GAD, HAD, and HAG 

are REAL words.  :-)

Edwin