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?

The piggy contains $20$ coins, that is, $20$ spaces.
For the first space, there are $5$ possibilities: it can be a penny, a nickel, a dime, a quarter or a half dollar. For the second space we have the same possibilities because coins can be repeated. For 2 spaces we would have $5.5=5^2$ possibilities for combining coins so for $20$ spaces we would have $5^{20}$ possibilities.
