Question

Five cards are dealt from 52.

Answer 1

For reference,

$\dbinom nk=\dfrac{n!}{k!(n-k)!}$

a. $\dbinom{52}5=2,598,960$ - nothing special here, you're just choosing any 5 cards from the deck

b. $\dbinom{13}4\dbinom{39}1=27,885$ - 13 hearts to choose from, and 39 of any other suit

c. $\dbinom{40}5=658,008$ - there are 12 face cards to omit from the count

d. $\dbinom{26}5=65,780$ - half the deck contains spades/clubs

e. $\dbinom{26}5=65,780$ - essentially the same situtation as (d)

f. $\dbinom{40}5\dbinom{12}0+\dbinom{40}4\dbinom{12}1+\dbinom{40}3\dbinom{12}2=2,406,768$ - either 0, 1, or 2 face cards are allowed

g. $\dbinom{13}2\dbinom{39}3+\dbinom{13}3\dbinom{39}2+\dbinom{13}4\dbinom{39}1+\dbinom{13}5\dbinom{39}0=953,940$ - similar to (f)