In how many ways can the letter oh the word "TRAINGLE" be arranged so that no two vowels are next to each other?

Question

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Answer 1

Ignoring for a moment the requirement that no two vowels are together there are 8! (=8*7*6...*2*1=40320) ways of arranging the 8 different letters. The reason for this is that, we have a choice of 8 letters for the first position; that leaves 7 for the second position, making 56 for the first two positions; then we have a choice of 6 letters for the third position, and so on. So, in all we have 8*7*6*5*4*3*2*1=8!=40320 possible arrangements of letters.

Similarly, there are 5! (=120) ways of arranging 5 consonants and 3! (=6) ways of arranging 3 vowels.

Now we have to find out how many have vowels together. 

TRIANGLE has 3 vowels and 5 consonants, and there are 8!/(5!)(3!))=only 56 ways of mixing the vowels and consonants (the reason for dividing by 5! and 3! is to reduce the number 40320 to compensate for all the ways we can arrange 5 different consonants and 3 different vowels, so that we treat all consonants as one and all vowels as one):

VVVCCCCC VVCCCCCV VVCCCCVC VVCCCVCC VVCCVCCC VVCVCCCC (6)

VCVVCCCC VCVCCCCV VCVCCCVC VCVCCVCC VCVCVCCC (5)

VCCVVCCC VCCVCCCV VCCVCCVC VCCVCVCC (4)

VCCCVVCC VCCCVCCV VCCCVCVC (3)11

VCCCCVVC VCCCCVCV (2)

VCCCCCVV (1) Total=21, 10 with vowels separated;

CVVVCCCC CVVCCCCV CVVCCCVC CVVCCVCC CVVCVCCC (5)

CVCVVCCC CVCVCCCV CVCVCCVC CVCVCVCC (4)

CVCCVVCC CVCCVCCV CVCCVCVC (3)

CVCCCVVC CVCCCVCV (2)

CVCCCCVV (1) Total=15, 6 with vowels separated;

CCVVVCCC CCVVCCCV CCVVCCVC CCVVCVCC (4)

CCVCVVCC CCVCVCCV CCVCVCVC (3)

CCVCCVVC CCVCCVCV (2)

CCVCCCVV (1) Total=10, 3 with vowels separated;

CCCVVVCC CCCVVCCV CCCVVCVC (3)

CCCVCVVC CCCVCVCV (2)

CCCVCCVV (1) Total=6, 1 with vowels separated;

CCCCVVVC CCCCVVCV (2)

CCCCVCVV (1) Total=3, none with vowels separated;

CCCCCVVV (1) Total=1, none with vowels separated.

From these you can see a pattern and we can also count how many have vowels (V's) together: 36; and 20 where the vowels are separated. So out of 40320 possible arrangements 20/56=5/14 of the total number of arrangements have the vowels separated: 5/14*40320=14400.

Another way to do the calculation is to take the 20 required vowel-consonant arrangements and multiply by the numbers of ways of arranging the vowels and consonants: 20*120*6=14400.

Answer 2

Here there are a total of eight choices for the first letter, seven for the second, six for the third, and so on. By the multiplication principle we multiply for a total of 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 8! = 40,320 different ways.

Total number of arrangements have the vowels separated: 5/14 * 40320 = 14400.