How many 10 letter sequences are there using 5 vowels and 5 consonants? What is the pribability ome of these words has no consecutive pair of consonants?
For the first part I reasoned that each consanta amd vowel could be chosen repeatedly thus: $(21^5)(5^5)$
Now the second part I have been finding tricky:
to rephrase the question I asked: how many different arrangements where a vowel amd consonant alternate such as "vcvcvcvcvc" where v is vowel and c is consonant. Using this idea as the template i initially thought something alomg the lines of $(5)(21)(5)(21)….$ and then multiply by 2 to account for the fact that either my vowel or consonant could be first. But doing this would make my numerator larger than my denominator with a value of : $(21^5)(5^5)(2)$
obviously something wemt wrong in my decomposition.
Suggestions?
Best Answer
HINT: For the first part of the question, remember that you also need to account for the fact that order matters in the sequence.
Fixing this should also fix your problem with the second part of the question.