this is a question I had on my midterm, and I can't seem to be sure what the answer is and our professor did not post the solutions, therefore I cannot make sure I got it right (or wrong).
How many length 10 words can be written in the alphabet {a,b,c,d,e} with exactly 5 vowels and 5 consonants?
My thought process has been the following.
1) Permutations with repetitions allowed on vowels and consonants. $2^5, 3^5 $ respectively.
2) Total combinations with 10 objects and 5 combinations. C(10, 5)
Resulting into:
$ {{2^{5}} * {3^{5}} * {\frac{10!}{5!5!}} = 1959552} $
Thank you for the clarification.
Yes, that looks correct; like you said, there are $10C5$ ways of choosing the 5 spots for the vowels (which means that the other 5 spots are already chosen), and for each of these 5-ples, you can choose the vowels in $2^5$ ways and the consonants in $3^5$ ways.