Chapter 26

Combinatorics

Many GRE problems are, ultimately, just about counting things. Although counting may seem to be a simple concept, problems about counting can be complex. In fact, counting problems have given rise to a whole subfield of mathematics: combinatorics, which is essentially “advanced counting.” This chapter presents the fundamentals of combinatorics that are essential on the GRE.

In combinatorics, you are often counting the number of possibilities, such as: How many different ways you can arrange things? For instance, you might ask the following:

• A restaurant menu features 5 appetizers, 6 entrées, and 3 desserts. If a dinner special consists of 1 appetizer, 1 entrée, and 1 dessert, how many different dinner specials are possible?
• Four people sit down in 4 fixed chairs lined up in a row. How many different seating arrangements are possible?
• If there are 7 people in a room, but only 3 chairs in a row, how many different seating arrangements are possible?
• If a group of 3 people is to be chosen from 7 people in a room, how many different groups are possible?