A boy has the following seven coins in his pocket: $2$ pennies, $2$ nickels, $2$ dimes, and $1$ quarter. He takes out two coins, records the sum of their values, and then puts them back with the other coins. He continues to take out two coins, record the sum of their values, and put them back. How many different sums can he record at most?
Ok. Let's list out each sum that is possible. We could have two pennies, which would be $0.02.$ We could have two nickels, which would be $0.10.$ We could have two dimes, which would be $0.20.$ We could have one penny and one nickel, which would be $0.06.$ We could have one penny and one dime, which would be $0.12.$ We could have one penny and one quarter, which would be $0.31.$ We could have one nickel and one dime, which would be $0.15.$ We could have one dime and one quarter, which would be $0.35.$ And those are all the possible sums. Right, so the answer is 9.

