Could someone confirm my combinatorics solutions for this question?
Part 1
How many eight letter strings of letters contain exactly two vowels?
Solution:
- 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: $C(8,2) \cdot 5^2 \cdot 21^6$
Part 2
How many eight letter strings of letters contain exactly two vowels if the two vowels cannot be adjacent?
Solution:
- 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: $21^6 \cdot C(7,2) \cdot 5^2$
Best Answer
I agree with the first one completely.
I'm not sure what the Separation Technique is.
I would count the positions for the vowels that are non-adjacent. We have 7 pairs of places that are adjacent: 12, 23,..., 78. So we have $\binom{8}{2} - 7$ many position pairs for the vowels.
The rest of the counting is the same: $5^2 \cdot 21^6$.