How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a piggy bank contain if it has 20 coins in it? How many different combinations of pennies, nickels,
dimes, quarters, and half dollars can a piggy bank contain if it has 20 coins in it?

If the order matters, we can use the product rule to show each of the $20$ coins can be chosen in $5$ ways, for a total of $5^{20}$ ways, assuming you have enough of each type of coins.
If the order does not matter (which in the case it probably does not) you have to use an $r$-combination. The formula is $C(n+r-1, r)$. Here $n = 5$ and $r = 20$. So we get $C(24,20)$.
