Mastering Permutation Combination

 

Combinatorics is not difficult, it's different. We deal it very similar to other branches of mathematics, i.e. use the equation/formulae to get an answer. And this fails. in some sense, every problem in combinatorics requires a different formula. My conclusion is "you can not master combinatorics by remembering a few set of formulae." 

Combinatorics can be treated as SCIENTIFIC COUNTING. 

To begin with, combinatorics the most basic part is the fundamental theorem of counting, create your own logic of connecting information given in the problem, and practice a lot more problem.

Every good combinatorics problem requires a new road map, create it.

The biggest confusion for beginners is in fact when to apply Permutation and when to apply Combination. Here is an easy way that can clear you concept: With permutations, every little detail matters. Ali, Bobby and Charl is different from Charl, Bobby, and Ali. Combinations, on the other hand, are pretty easy going. The details of order do not matter. Ali, Bobby, and Charl is the same as Charl, Bobby, and Ali. And permutations are for lists (order matters) and combinations are for groups (order doesn’t matter). 

 To get the answer to What is Combinatorics? I would like to take you to the beginning of the subject.

Combinatorics is a young field of mathematics, starting to be an independent branch only in the 20th century. However, combinatorial methods and problems have been around us ever since. Many combinatorial problems look entertaining and one can easily say that roots of combinatorics lie in mathematical creativity and games.

Nonetheless, this field has grown to be of great importance in our today’s world, not only because of its use for other fields like physical sciences, social sciences, biological sciences, information theory, and computer science. Combinatorics is concerned with: Arrangements of elements in a set into patterns satisfying specific rules.

Now the more interesting question is, How to excel in this beautiful and so useful subject?

Combinatorics is built on, two basic rules. We call these basic rules that ease out counting burden, the Fundamental Principle of Counting. If we want to do counting smartly, we have to work upon these fundamental rules first. To do so, let’s start with a very interesting counting problem.

To introduce the principles, let us take an example of a car brand that sells the following three categories of cars,

And six different models of each say LXi, VXi, ZXi and LDi, VDi, ZDi. Now can you count, how many different types of cars does the car brand sell?

Before moving ahead, think about it for a moment and try to find out the answer yourself.

Well, let us create a list of all the possible cases one by one and count all these cases.

Lets us talk about hatchback cars first,

There are 6 different Hatchback cars by brand viz hatchback of LXi, VXi, ZXi and LDi, VDi, ZDi.

There are 6 different sedan cars by brand viz sedan of LXi, VXi, ZXi and LDi, VDi, ZDi.

And there are 6 different suv cars by brand viz suv of LXi, VXi, ZXi and LDi, VDi, ZDi

Finally, if we count, a total of 6+6+6 = 18 types of cars are available with the brand. All right!

Here is how we can reach that number by a simple calculation.

Since there are 3 car categories (hatchback, sedan & SUV), and each of these comes in 6 different models, the total number of types of cars can be obtained by multiplying the number of categories (3) with the number of models available for each category (6), Which is 3 x 6 = 18

Now I am going to make this counting a little more complicated. Suppose each of the above models was available in 4 different colors.

How many different cars do you think are available now?

Let us do the counting again. Previously we had 18 different cars, now each car is available in four color options.

Since each of the cars is available in 4 colors, and there are 18 different types of cars (as we calculated above), we have 18 black cars +18 blue cars +18 orange cars +18 white cars = 18 x 4 = 72 different cars.

And this number can be obtained by multiplying together, the number of categories of cars (3), the number of models for each category (4), and the number of colors available for each model (5) i.e. 3x 4x 5= 60.

Congratulations, you have built two fundamental theorems of counting. Let me now state the same little mathematically.

If a task T can be divided into subtasks T1 and T2, which can be completed in m ways and n ways respectively, and T will be completed by completing either T1 or T2, then the number of ways of completing T will be m + n. We call this Addition Principle.

If a task T can be divided into subtasks T1 and T2, which can be completed in m ways and n ways respectively, and T will be completed by completing both T1 and T2, then the number of ways of completing T will be m.n. We call this Multiplication Principle.

Hope you enjoyed learning combinatorics. 

Happy Math Learning Visually