a)First letter a consonant, did I read well?

Suppose you put the H in the first place; you get

H _ _ _ _ _ that means you have 5 letters to put in 5 places; thatīs a permutation of 5 elements:

Doing the same with the other consonants it means that you have ways.

b) _ vowel _ _ vowel _

and you only have two possibilities:

_ A _ _ E _ (permutation of 4 elements)

or

_ E _ _ A _ (permutation of 4 elements)

that leads to

Iīm assuming there are no repeated letters on an arrangement.