## Vigornii Alphabet

In the Vigornii alphabet there are $8$ consonants and $4$ vowels. How many different arrangements of $5$ letters can be made if exactly $2$ vowels must be used and no repetition of letters is allowed?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
1. Choose $2$ places out of $5$ places to put the $2$ vowels
$\binom{5}{2}=10$
2. Put $2$ vowels out of $4$ vowels in the $2$ chosen places
$10\times 4\times 3=120$
3. In the remaining $3$ places put $3$ consonants out of $8$ consonants
$120\times 8\times 7\times 6=40320$
There are $40320$ different arrangements.

Alternative solution
1. Choose $2$ vowels out of $4$ vowels
$\binom{4}{2}=6$
2. Choose $3$ consonants out of $8$ consonants
$\binom{8}{3}=56$
3. How many ways can we form a word with $2$ vowels and $3$ consonants?
$6\times 56=336$
4. For each of the $336$ five-letter words how many different ways can we arrange the $5$ letters?
$336\times 5!=40320$