SOLUTION: How many permutations can be made from the letters of the word "WHITE" with vowels and consonants alternating?

Question 997223: How many permutations can be made from the letters of the word "WHITE" with vowels and consonants alternating?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i believe it's going to be 2! * 3! = 2 * 6 = 12.

the consonants have to be on the outside and the vowels have to be on the inside.

why?

because if a voowel is on the outside than 2 consonants have to be together.

let c = consonant
let v = vowel

cvcvc is ok
vcvcc is not ok
ccvcv is not ok.

so vowels have to be in the middle.

so slots 1, 3, and 5 have to be consonants.
and slots 2 and 4 have to be vowels.

within the 3 consonent slots, you can lay the consonents out in 3! = 6 ways.

within the vowel slots, you can lay the vowels out in 2! = 2 ways.

the total number of ways is therefore 3! * 2! = 6 * 2 = 12 ways.

since the number of vowels and consants is small, we can actually lay them out so you can say how it works.

the consonents are wht
the vowels are ie

the different number of ways the vowels can be layed out are:

ie
ei

the different number of ways the consonents can be layed out are:

wht
wth
hwt
htw
twh
thw

now, for wht, you can intersperse ie two different ways.

you get:

wihet
wehit

for wth, you can intersperse ie two different ways.

you get:

witeh
wetih

you can do the same for all 6 possible combinations of the consonents, so the total number of ways is 6 * 2 = 12.

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