How many five letter words can be formed from the English alphabet that contain $2$ different consonants and $3$ different vowels?

Question

The English alphabet has $26$ letters, of which $5$ are vowels.

(a) How many five letter "words" containing $2$ different consonants and $3$ different vowels can be formed?

(b) How many of these "words" begin with "b" and end in "a"?

I have done part a $21C2 \cdot 5C3 \cdot 5! = 252000$. Not sure how to solve part b.

Answer 1

How many five letter words can be formed from the English alphabet that contain $2$ different consonants and $3$ different vowels?

Your answer is correct.

How many of these words begin with $b$ and end with $a$?

We have to choose one of the other $20$ consonants, two of the other four vowels, and then arrange the three chosen letters in the three spaces between $b$ and $a$.

$$\binom{20}{1}\binom{4}{2}3!$$

Answer 2

20C1*4C2*3! Fix the positions of b in the beginning and a at the end. Now you have 20 consonants and 4 vowels and 3 positions to fill.