Discover the Number of 8 Letter Words with 3 Vowels and Non-Repeating Consonants

[Miike012]

How many 8 letter words have 3 vowels constants do not repeat.

Is it
5^3*26*25*24*23*22 ?

 

[jedishrfu]

How many 8 letter words have 3 vowels constants do not repeat.

Is it
5^3*26*25*24*23*22 ?

Don't you have to also factor in order too:

act vs cat vs some nonsense words like tca, cta ...

cafe vs face

 

[Miike012]

Another question how do I know when to use the combination equation or permutation equation?
For instance given the example which is posted above are the following strings the same?

A E I M N P Q R
and
E A I M N P Q R
They both contain the same three vowels and the only thing I changed was the placement of the first and second terms

 

[Miike012]

Don't you have to also factor in order too:

act vs cat vs some nonsense words like tca, cta ...

cafe vs face

so I should add 8! ?

 

[Miike012]

Don't you have to also factor in order too:

act vs cat vs some nonsense words like tca, cta ...

cafe vs face

I'm not sure that would work, for instance


A₁ B A₂
and
A₂ B A₁

are the same.

I almost feel like I should add (8 choose 3) instead of 8!

 

[jedishrfu]

I'm not sure that would work, for instance


A₁ B A₂
and
A₂ B A₁

are the same.

I almost feel like I should add (8 choose 3) instead of 8!

I think your consonants are unique but your vowels might repeat so you'd also have to factor that in too.

 

[haruspex]

so I should add 8! ?

You mean multiply by 8!?
No, your original formula already has some ordering in it.
For the vowels you have 5³, so you've already counted all vowel orders.
Similarly, for the consonants, you used ²⁶P₅, not ²⁶C₅, so you have those ordered.
All that you are missing is integrating the vowels with the consonants. How many ways are there of doing that?

 

[vela]

Remember that the vowels are part of the 26 letters. There are only 21 consonants.

 

[jedishrfu]

Remember that the vowels are part of the 26 letters. There are only 21 consonants.

I missed that, thanks.

 

[haruspex]

Remember that the vowels are part of the 26 letters. There are only 21 consonants.

How true