# Permutations and Combinations | Counting

##### Permutations and Combinations Counting :The formula for N C R is ‘N factorial, divided by R’ factorial times ‘N minus R’ factorial.

What is this? This is the formula for combinations, but it is not the best way to understand it ! And what is NPR. It’s N factorial over **‘N minus R’ factorial**.

And this again, is not the best way to understand permutations! In many books, and article you would be asked to use this formula, if we have to select our things out of N.

And we would be asked to use this formula if we have to arrange our things out of n.

I can assure you that if you understand just these two formulae, you will never really understand permutations and combinations.

There’s only one thing you should understand well if you wish to master this topic! And that’s counting !

If you’re able to count well, then the topic of permutations and combinations will be a walk in the park! Of course, the counting will not be as simple as **counting** on your fingers.

It’s a little more advanced. But don’t worry. Counting is easy! Let’s say we have three pens, and two markers.

Every item is distinct. No two items are the same. Here’s your first question based on counting. In how many ways can we pick any one items from these five? Understand the question well.

In how many ways, can we pick any one item from three pens and two markers? In how many ways do you think we can do this? Look at it logically.

We can either pick the first pen. We can call it P1 or the second pen or maybe the third pen. Or maybe we pick a marker.

The first one or the second one! We kept saying OR. This or this or this and so on. You can see that there are five ways in which we can pick any one item.

And how do we get a five? Three ways in which a pen can be picked. Plus two ways in which your marker can be picked .Five ways in all. This is the first rule of counting .

Or !

Or, always means addition.

First Pen or the second pen or the third. Three ways in which a pen can be picked. And similarly, two ways in which your marker can be picked. Or always means addition. Now let me ask you another one. In how many ways can we pick one pen and one marker?

one Pen and one marker! Tell me what you think? We have to pick a pen and a marker. In how many ways can we pick a pen?

We pick either the first or the second or the third? Now having picked one pen we need to pick a marker. Maybe after picking the first pen, we pick the first marker.

Or May be after picking the first pen, we pick the second marker. So one way is p 1 m 1, and another is p 1 m 2.

Or Maybe we pick p 2 and then m 1 or p 2 and m 2 .

In all we can see that there are six ways in which we can pick a pen and a marker.

Which are the six ways?

p 1 m 1,

p 1 m 2,

p 2 m 1, p 2 m 2, p 3 m 1 and p 3 m 2 .

And how did we get a 6? 3 ways times two ways. Multiplication! ‘And’ is another rule of counting, it always means multiplication. One pen and one marker ‘three ways’ multiplied by ‘two ways’. One pen or one marker three ways plus two ways.

If you understand these two rules of counting, trust me! You will be able to solve most problems based on permutations and combinations! And this is nothing new! I am sure you have used this many times in life.

Look at this picture! How many circles do you see? I doubt you counted each circle. Instead of counting each circle,

you would have noticed that there are six columns and four rows. Because it’s AND the number of Circles will be 6 x 4. 24 and all.

Don’t forget the two basic Rules. AND is multiplication OR is addition. The topic of permutations

and combinations is not liked by many students and maybe you as well. So we have covered as many topics and examples as we can.

And I can assure you that if you watch all our videos based on this topic, you will master the concepts! So take some time, go through our sessions, and you will realize how simple this topic is!