Here is the question:
We consider the Latin alphabet with $26$ letters, of which $5$ are vowels. How many words can we form that start with b and contain c and have $2$ vowels and $3$ consonants in total?
We already have $2$ consonants so we need $1$ consonant and $3$ vowels, we have $4$ cases:
case one: b c _ _ _ case two: b _ c _ _ case three: b _ _ c _ case four: b _ _ _ c
In each case the vowels can be arranged in $C(3,2)$ ways so in total $4 \cdot C(3,2)$ ways and the consonants can be arranged in $C(3,1)$ ways so in total $4 \cdot C(3,1)$
Final answer: $4 \cdot C(3,2)+4 \cdot C(3,1)$