You have in your pocket a half-dollar, a quarter, a dime, and a nickel. You are at a flea market, staring at a table of various small items. The clerk shows you several of them, each of which has a different price, and any one of which can be purchased with one or more of the coins in your pocket without your receiving any change. What is the largest number of items the clerk could show you?
Checking all the combinations of 1, 2, 3, and 4 coins, we find these possible prices:
One coin: 5, 10, 25, 50
Two coins: 15, 30, 55, 35, 60, 75
Three coins: 40, 65, 80, 85
Four coins: 90
So there could be as many as 15 different items altogether.