In grouping problems, you make complete groups of items, drawing these items out of a larger pool. The goal is usually to maximize or minimize some quantity, such as the number of complete groups or the number of leftover items that do not fit into complete groups. As such, these problems are often really a special case of optimization problems. One approach is to determine the limiting factor on the number of complete groups. That is, if you need different types of items for a complete group, figure out how many groups you can make with each item, ignoring the other types (as if you had unlimited quantities of those other items). Then compare your results. For example:
Orange Computers is breaking up its conference attendees into groups. Each group must have exactly one person from Division A, two people from Division B, and three people from Division C. There are 20 people from Division A, 30 people from Division B, and 40 people from Division C at the conference. What is the smallest number of people who will NOT be able to be assigned to a group?
The first step is to find out how many groups you can make with the people from each division separately, ignoring the other divisions. There are enough Division A people for 20 groups, but only enough Division B people for 15 groups (= 30 people ÷ 2 people per group). As for Division C, there are only enough people for 13 groups, because 40 people ÷ 3 people per group = 13 groups, plus one person left over. So the limiting factor is Division C: only 13 complete groups can be formed. These 13 groups will take up 13 Division A people (leaving 20 − 13 = 7 left over) and 26 Division B people (leaving 30 − 26 = 4 left over). Together with the 1 Division C person left over, there are 1 + 4 + 7 = 12 people will be left over in total.
For some grouping problems, you may want to think about the most or least evenly distributed arrangements of the items. That is, assign items to groups as evenly (or unevenly) as possible to create extreme cases.
A salad dressing requires oil, vinegar, and water in the ratio 2:1:3. If Oliver has 1 cup of oil, 1/3 cup of vinegar, and 2 cups of water, what is the maximum number of cups of dressing that he can mix?