It is the different arrangements of a given number of elements taken one by one, or some, or all at a time

But Before That Just Look at A very Important Concept without Which You can’t solve a single Question of permutation/combination or probability.

And that Factorial Notation.

It’s represented by (!) and it is read as Factorial.

So if i write 5! it will be read as Five Factorial.

And what it means ? It means to simply multiply all the numbers in decreasing order till 1.

Like if i write 6! it means 6*5*4*3*2*1 = 720

Or 7! = 7*6*5*4*3*2*1   = 5040

For Fast Calculation You all must learn the value of factorial till 10.

Just Learn these values

1! = 1

2! = 2

3!= 6

4! = 24

5! = 120

6! = 720

7! = 5040

8! = 40320

9! = 362880

10! = 3628800

 

 

So how to do permutation?

Suppose there are 3 words ABC and if it’s asked How many ways these three can be arranged then all you or What are the no. of permutations Possible. Then all you have to do is Arrange this things in as many ways it’s Possible.

Let’s try to arrange them now. SO There is ABC, ACB, BAC, BCA, CAB, CBA Are there any more ways these can be arranged ?try it ? No These are the all possible arrangements. So The answer to the above Question will be 6. That is ABC can be arranges in different ways.

Now there were only 3 alphabets What if there were more like You have to Arrange ABCDEFGHI. Now for 3 alphabets it was easy you easily arranged them But Arranging these 9 letters will take you days and even then you will not be able to get a certain answer.

So what we should do here. Just apply a  simple formula for that.

And Formula is like this.

N Different things can be arranged in N! ways.

So in above Question there were 9 alphabets so the no. of possibele arrangements will be 9! = 362880.

So that was out basic concept Now let’s move on to another basic concept.

So in the above questions It was Asked in how many ways ABCDEFGHI Can be arranged. In this question they were asking the possible arrangements of all the 9 Alphabets, They can also Ask In how many ways 4 alphabets from above 9 alphabets can be arranged.

In such type of Questions there is another formula Which is very veryvery important because it will be used in almost every question.

So the formula is Out of n things r things can be arranged in nPr ways. and

nPr = n!/(n-r)!

So in the above Question it is asked that in how many ways 4 alphabets from the total 9 alphabets can be arranged.

So apply the formula nPr = 9P4 = 9!/(9-4)! = 9*8*7*6*5*4*3*2*1/5*4*3*2*1 =9*8*7*6 = 3024.

Now there is a trick to easily calculate nPr by which you won’t have to do any division work.

Like if it say 9P3 then you just have to multiply Starting from 9 in decreasing order till the next 2 digit i.e 9P3 = 9*8*7. Why we multiply till 7 only ?that is because the value of r is 3 and total multiplication should contain the value of r.

Another example if it 7P2 then you will just do 7*6[ 2 number because r = 2 ok]

if it’s 7P4 then the answer will be 7*6*5*3[ 4 no. because value of r=4]

So If it’s 10P5 then the value will be 10*9*8*7*6 [ 5 digit because value of r = 5]

Actually there are infinite cases in Permutation and Combination 100’s of different type of question can be formed So i will only discuss the cases that are important for the exam, And if you have any problem in any other case then you can ask me personally.

 

 

Case – 1 Simple Arrangement Case well all words are unique.

By UNIQUE itmean all alphabets are different

In how many ways the letters of the word ROCKET can be arrnged.

very Simple just count the no. of words in ROCKET that will be 6

So number of arrangements will be n! that will be 6!

 

 

CASE – 2 Arrangement When All the words are not UNIQUE.

That means some words are repeated.

Like No. of possible arrangements of word TITANIC

Now In this case you Just have to find the total possible ways first without even thinking about Repeated words and then after that You will divide that with the numbers of times a Word is repeated.

So in the above Question Total alphabets = T = 2, I = 2. A= 1 C =1 N = 1Total 7 So Permutations will be 7! and Now you will divide It by No. of times A word is repeated SO T is repeated 2 times and I is repeated two times So divde 7! by these 2. So final Answer will be 7!/(2!*2!)

 

 

Let’s See another Example. In how many ways the letters of the word RUNNING Can be arranged.

So total no. of alphabets in the above Words = 7

No. of words that are repeated = N = 3 times repeated.

So the solution will be Total permutation divided by no. of times a word is repeated and that will be 7!/3! that will be your answer.

 

 

Case 3 – Arrangement Some Words are always together and Some Words and Never together.

No of possible arrangements of the words LAYERING When Vowels are always together.

In this case what we do Is we consider the no. of Vowels as 1 single alphabet That [AEI] is a one single alphabet In that way they will always be together and the rest words are LYRNG.

So the total no. of alphabets will be 6 ? Why 5 Alphabets are LYRNG and [AEI] is Onealhpabet remember so The total alphabet will be 6

And no. of possible arrangements will be 6!

But but the question is not complete yet [AEI] Though considered as 1 alphabet but stil the words AEI can change places within itself Like AEI it also can be AIE or EIA. So there are 3 words so no. of total arrangements that they can do within itself will be 3!

So our final answer will be 6!*3![ that is because 6! is the no. of possible ways when AEI are together and And multiplied by 3! because AEI can change places within themselves in 3! possible Ways]

 

 

If it was asked that VOWELS in LAYERING are never together that what we will do ?

This Question can’t be solved directly.

In order to solve this We will have to FIND the total no. of arrangements of the word LAYERING and then Subtract the no. of arrangemnts in which AEI are Always together.

So no. of possible arrangements of LAYERING will be 8!

And We already Solved that when AEI are always together the no. of possible ways are 6!*3!

So no. of possible ways when AEI are never together will be 8! – 6!*3!

Now We should move on to the next Topic That Is Combination. Now you know that Permutation means Arrangement or no. of possible ways A thing can be arranged.

What is the meaning of Combination.

Combination is a simple act of Choosing or Selection.

Like When it is asked What are no. of possible ways Word TITAN can be arrange You have to find The Permutation.

But if it is asked what are no. of possible ways You can Select 2 alphabet from the word TITAN, It means you have to find Combination.

The act of selection or Choosing is called COMBINATION.

Now you all must know what is nPr so it’s time to move towards nCr

Like nPr = n!/(n-r)!

 

 

nCr is somewhat simillar but that is just an extra r! in the denominator

So nCr = n!/[(n-r)!*r!]

nCr means r things has to be selected out of n things.

Like in the above Question No. of possible ways 2 alphabets can be selected from the word TITAN

So total no. of alphabets n = 5

  1. of alphabets which we have to select r = 2

So the answer will be 5C2 = 5!/(5-2)!*2! = 5!/3!*2! = 5*4/2*1 = 10

 

 

Now you have to calculating nPr in a simple way Just like that we can also calculate nCr in a simple way All you have to do is Follow the method of nPr and In division you have to also multiply in increasing order from 1

Like 6C3 = 6*5*4/1*2*3

And 9C2 = 9*8/1*2

and 10C4 = 10*9*8*7/1*2*3*4

7C5 = 7*6*5*4*3/1*2*3*4*5