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The location of the vowels can be chosen in ways.
For each choice of locations, the locations can be filled with vowels in ways.
Four empty locations can be filled with consonants in ways.
The total number of strings is
The number of strings which contain exactly vowels is
Exactly 1vowel: The location of the vowel can be chosen in ways.
For each choice of location, the location can be filled with a vowel in ways.
That leaves empty locations, which can be filled with consonants in ways
The number of strings which contain at least two vowels is