Question

Let $C$ be the set of 6 consonants $\{b,c,d,f,g,h\}$ and $V$ be a set of 5 vowels $\{a,e,i,o,u\}$ and $W$ be the set of seven letter words that can be formed with these 11 letters using both the following rules.
(a) The vowels and consonants in the word must alternate.
(b) No letter can be used more than once in a single word.
If the number of words in the set $W$ are $10K$ find $K$.

Answer 1

Answer: 3600

$\text{ Consonant }⟶b,c,d,f,g,h(6)$
Vowels $⟶$ a, e, i, o, u (5)

Case I: If word begins with consonants
$\text{ then }\left({}^6{C}_4\times 4!\right)\times \left({}^5{C}_3\times 3!\right)=360\times 60=21600$
Case II: If word begins with vowels
$\left({}^5{C}_4\times 4!\right)\times \left({}^6{C}_3-3!\right)=120\times 120=14400$
Total $=36000\Rightarrow 10K=36000\Rightarrow K=3600$