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.