Introduction: Combinations and calculations

In books designed for beginning algebra students – and, no, this is not an algebra text with the wrong cover – questions of the following sort commonly are posed: If John can mow the lawn in forty-five minutes and Mary can mow the lawn in thirty minutes, how long will it take John and Mary to mow the lawn together? My first instinct when I encountered such problems was not to figure out what fraction of the lawn each mowed in a minute so that I could add their contributions and then work out the length of time needed for the team to complete its work. Instead, I was inclined to ask whether John and Mary each had a lawnmower, whether they got along well or tended to compete with one another, and, if they were competitive, whether it was a good-natured competition (where each tried to be better than the other) or a nastier version (where they would sabotage one another's efforts).

Although that instinct was not always applauded by my teacher, it poses useful questions. There are times when two people possess complementary skills and can be induced to use them to greater advantage than a mere sum of their individual attainments would suggest. And there are instances in which individuals are better left on their own, when either would do better than the two in combination.