Counting is one of the first skills any student learns in school and it's a skill that you need practically every day. But there's also a complex and beautiful side to counting: a mathematical field called Combinatorics that is an active field of research and a source of many challenging puzzles and problems.
As you begin to learn the essential techniques that apply to Combinatorics, you'll start by learning how to rearrange and group objects to create symmetrical arrangements that are easier to count. And the more combinatorics problems you work on, the more your toolbox of counting techniques will expand from there!
Here are some tips to get you started:
- When given a scenario or pattern in a problem, experiment with it. Don’t be afraid to try many different things, to move objects around and to group or combine them in different ways.
- Symmetry is at the heart of many combinatorics puzzles. Look for a symmetrical way to rearrange or group objects that you're counting. Symmetry makes large sets of objects much easier to count.
- Wishful thinking is actually a very useful problem-solving strategy. Ask yourself "what pattern, if it were completed here, would make this problem easy to solve?" Then, if you're close to the pattern, ignore the difference temporarily and factor it back in at the end.
- As you practice with many problems, you'll gradually build up a tool kit of counting strategies.
- If you get stumped on a particular problem, look at the solution and be sure to study the technique so that you can add it to your toolkit.
In Combinatorics, you will learn how to strategically solve a tremendous variety of problems. The methods you'll learn include:
- Checking Cases: Taking an organized approach and splitting difficult problems into parts and cases is an essential skill to master.
- Using the Rule of Sum and Rule of Product: If you group the objects that you're counting wisely, the symmetry within the set of your objects and groups can let you use addition and multiplication as shortcuts to count.
- Counting Permutations: Counting the rearrangements of sets of objects.
- Counting Combinations: Counting the number of ways to partition a set of objects into groups.
- Principle of Inclusion and Exclusion (PIE): If the objects being counted are part of a larger context, there are many subtle and clever ways to count a set of objects indirectly, if it's too difficult to count directly.
Additionally, other mathematical fields including Geometry, Algebra, and Number Theory also have counting-related questions. For example: How often do the graphs of two specific functions intersect? How many prime numbers are there between 2 and 4 vs. 4 and 8 vs. 8 and 16...? How many different paths are there between two destinations in a transportation network? Counting challenges such as these can be extremely difficult, nothing like counting "\(1, 2, 3..\)."