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The former, the optimization problem, should be very familiar. It is a problem that cashiers solve when (for example) you hand them a five-dollar bill for a purchase of $3.44, and they have to find a way to give you back $1.56 in ''change'', preferably with as few coins as possible.<ref>The Canadians have a $2.00 coin, a $1.00 coin, a $0.25 coin, a $0.10 coin, a $0.05 coin, and a $0.01 coin. The solution to this problem in Canadian currency is one $1.00 coin, two $0.25 coins, one $0.05 coin, and one $0.01 coin.</ref> It is a well-studied problem that is extensively treated in the literature. The counting problem is somewhat more exotic; it arises mostly as a mathematical curiosity, often in the form of the puzzle: "How many ways are there to make change for a dollar?"<ref>The denominations in circulation in the United States are the same as those in Canada, except for the absence of a $2.00 coin and the presence of a $0.50 coin. In United States currency, the answer is 293 if the trivial solution consisting of a single one-dollar coin is counted, or 292 if it is not.</ref> It is also not often encountered in the literature, but it is discussed here because it occasionally appears in algorithmic programming competitions. | The former, the optimization problem, should be very familiar. It is a problem that cashiers solve when (for example) you hand them a five-dollar bill for a purchase of $3.44, and they have to find a way to give you back $1.56 in ''change'', preferably with as few coins as possible.<ref>The Canadians have a $2.00 coin, a $1.00 coin, a $0.25 coin, a $0.10 coin, a $0.05 coin, and a $0.01 coin. The solution to this problem in Canadian currency is one $1.00 coin, two $0.25 coins, one $0.05 coin, and one $0.01 coin.</ref> It is a well-studied problem that is extensively treated in the literature. The counting problem is somewhat more exotic; it arises mostly as a mathematical curiosity, often in the form of the puzzle: "How many ways are there to make change for a dollar?"<ref>The denominations in circulation in the United States are the same as those in Canada, except for the absence of a $2.00 coin and the presence of a $0.50 coin. In United States currency, the answer is 293 if the trivial solution consisting of a single one-dollar coin is counted, or 292 if it is not.</ref> It is also not often encountered in the literature, but it is discussed here because it occasionally appears in algorithmic programming competitions. | ||
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==Greedy algorithm== | ==Greedy algorithm== |