Answer
a. 4
b. $1020$
c. 13
d. $3744$
Work Step by Step
a. Royal flush
4 (AKQJ10, 4 types)
b. Straight flush (not including a royal flush)
each position have 4 choices, 5 cards, $4^5-4=1020$
c. Four of a kind
13 (AAAA, KKKK, etc )
d. Full house
There are 13 choices for the three cards, the three cards come from a 4-card pool ($_4C_3=4$), total $13\times 4$
There are 12 choices for the pair which come from a 4-card pool ($_4C_2=6$), total $12\times 6$
Total for full house: $13\times 4\times12\times 6=3744$
