SOLUTION: How many ways can you arrange the letters of the word MAMMALIA provided no vowels are put together.

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Question 1156752: How many ways can you arrange the letters of the word MAMMALIA provided no vowels are put together.
Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


There are 4 vowels and 4 consonants; if the vowels can't be together, then the arrangement must have alternating vowels and consonants; either a vowel or a consonant can be first.

The 4 vowels are AAAI. The number of ways to arrange those vowels is .

The 4 consonants are LMMM. The number of ways to arrange those consonants is .

4 ways to arrange the vowels; 4 ways to arrange the consonants; and 2 ways to choose whether the first letter is a vowel or a consonant means 4*4*2=32 different arrangements.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
I'm afraid the other tutor's solution is incorrect.  But we're just human, and
all of us mess up sometimes.  I've messed up a number of times!

MAMMALIA

The vowels are A,A,I, and A.
The consonants are M,M,M, and L.

There are only 5 basic ways we can arrange them, vowel and consonant-wise 
so that no vowels are put together:

CVCVCVCV, VCVCVCVC, VCCVCVCV, VCVCCVCV, VCVCVCCV

There are 4 vowels, among which 3 A's are indistinguishable.
There are 4 consonants, among which 3 M's are indistinguishable.

So for each of those 5 there are 4!/3!=4 distinguishable ways to place the 4 vowels.

And there are 4!/3!=4 distinguishable ways to place the 4 consonants.

So there are 5*4*4=80 ways to place them.  Here are all 80, generated
by a computer program in alphabetical order.  It's fun trying to pronounce
them. My favorite is Number 73, MAMALIMA. If your MAMA was a LIMA, you'd be
a son of a bean. lol

 1.  ALAMAMIM
 2.  ALAMAMMI
 3.  ALAMIMAM
 4.  ALAMIMMA
 5.  ALAMMAMI
 6.  ALAMMIMA
 7.  ALIMAMAM
 8.  ALIMAMMA
 9.  ALIMMAMA
10.  ALMAMAMI
11.  ALMAMIMA
12.  ALMIMAMA
13.  AMALAMIM
14.  AMALAMMI
15.  AMALIMAM
16.  AMALIMMA
17.  AMALMAMI
18.  AMALMIMA
19.  AMAMALIM
20.  AMAMALMI
21.  AMAMAMIL
22.  AMAMAMLI
23.  AMAMILAM
24.  AMAMILMA
25.  AMAMIMAL
26.  AMAMIMLA
27.  AMAMLAMI
28.  AMAMLIMA
29.  AMAMMALI
30.  AMAMMILA
31.  AMILAMAM
32.  AMILAMMA
33.  AMILMAMA
34.  AMIMALAM
35.  AMIMALMA
36.  AMIMAMAL
37.  AMIMAMLA
38.  AMIMLAMA
39.  AMIMMALA
40.  AMLAMAMI
41.  AMLAMIMA
42.  AMLIMAMA
43.  AMMALAMI
44.  AMMALIMA
45.  AMMAMALI
46.  AMMAMILA
47.  AMMILAMA
48.  AMMIMALA
49.  ILAMAMAM
50.  ILAMAMMA
51.  ILAMMAMA
52.  ILMAMAMA
53.  IMALAMAM
54.  IMALAMMA
55.  IMALMAMA
56.  IMAMALAM
57.  IMAMALMA
58.  IMAMAMAL
59.  IMAMAMLA
60.  IMAMLAMA
61.  IMAMMALA
62.  IMLAMAMA
63.  IMMALAMA
64.  IMMAMALA
65.  LAMAMAMI
66.  LAMAMIMA
67.  LAMIMAMA
68.  LIMAMAMA
69.  MALAMAMI
70.  MALAMIMA
71.  MALIMAMA
72.  MAMALAMI
73.  MAMALIMA
74.  MAMAMALI
75.  MAMAMILA
76.  MAMILAMA
77.  MAMIMALA
78.  MILAMAMA
79.  MIMALAMA
80.  MIMAMALA

Edwin
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