For reference,
[tex]\dbinom nk=\dfrac{n!}{k!(n-k)!}[/tex]
a. [tex]\dbinom{52}5=2,598,960[/tex] - nothing special here, you're just choosing any 5 cards from the deck
b. [tex]\dbinom{13}4\dbinom{39}1=27,885[/tex] - 13 hearts to choose from, and 39 of any other suit
c. [tex]\dbinom{40}5=658,008[/tex] - there are 12 face cards to omit from the count
d. [tex]\dbinom{26}5=65,780[/tex] - half the deck contains spades/clubs
e. [tex]\dbinom{26}5=65,780[/tex] - essentially the same situtation as (d)
f. [tex]\dbinom{40}5\dbinom{12}0+\dbinom{40}4\dbinom{12}1+\dbinom{40}3\dbinom{12}2=2,406,768[/tex] - either 0, 1, or 2 face cards are allowed
g. [tex]\dbinom{13}2\dbinom{39}3+\dbinom{13}3\dbinom{39}2+\dbinom{13}4\dbinom{39}1+\dbinom{13}5\dbinom{39}0=953,940[/tex] - similar to (f)
