Could someone confirm my combinatorics solutions for this question?
How many eight letter strings of letters contain exactly two vowels?
- Choose two spots out of eight possible for the two vowels, order does not matter -- $C(8,2)$.
- Pick a vowel for each spot. There are two spots, five vowels in the alphabet and "no repeats" condition was not specified, so there are $5^2$ choices.
- Pick the remaining six consonants, which is $21^6$, since there are $21$ consonants and six spots left.
Answer: $C(8,2) \cdot 5^2 \cdot 21^6$
How many eight letter strings of letters contain exactly two vowels if the two vowels cannot be adjacent?
- Using the Separation Technique, space out and place the six possible consonants, creating seven possible positions for the two vowels -- $21^6$.
- Out of the seven spacer spots, pick two to be used for the two vowels -- $C(7,2)$.
- There are five choices per spot and "no repeats" restriction was not specified -- $5^2$.
Answer: $21^6 \cdot C(7,2) \cdot 5^2$