Nonempty set mapped to $\emptyset$ and vice-versa

Nonempty set mapped to $\emptyset$ and vice-versa

$(a)$ How many functions are there from a nonempty set $S$ into $\emptyset$?
$(b)$ How many functions are there from $\emptyset$ into an arbitrary set
$S$?
This question seems very simplistic but I don't know the answer. I think
for $(a)$ that there isn't a function that maps a set $S$ into a empty
set? For $(b)$ I assume it to be all function that map the empty set to an
arbitrary set since all sets contain the empty set?