How many 6-letter words that have either exactly 2 vowels or 4 vowels are there? (all lower case)
We can get around it as follows for the two vowel case:
Represent the word as six blank spots: _ _ _ _ _ _.
Choose two spots for the vowels: ${6 \choose 2}$.
Since order matters, we can fill those two spots in $5^2$ ways.
Since order matters, we can fill the remaining consonants in $21^4$ ways.
For a total of ${6 \choose 2} 5^2 21^4=72930375$ ways.
A similar argument applies to the four vowel case. I get a final total of $77064750$ different words with either exactly two or exactly four vowels.