Mathematics is life, not everyone knows this. Ever wondered why it's made compulsory when you pass through school? You can't escape it because you perform some arithmetics in every day.

I'll be dropping some tutorials on mathematics. And the first topic I would like to treat is Permutation.

Just sit down and relax as I work you through.

Now, what is a PERMUTATION?

PERMUTATION is the several possible ways in which a set or number of things can be ordered or arranged.

Example 1 - The number of arrangements of 3 objects A, B and C can be obtained from the tree diagram below.

The above tree diagram is showing permutations of the three letters.

There are six possible arrangements which are

ABC BAC CAB

ACB BCA CBA

A factorial notation is just the product of all consecutive integers starting from 1 to n.

In mathematics, the sign ! is known as factorial.

Thus n!=n * (n -1) * (n-2) .... 3 * 2 *1

Note:- This formula is used for n greater than 0.

These are examples to illustrate factorial notation.

               1!=1

               2!=2 * 1=2

               3!=3 *2 *1=6

The number of permutations of n distinct objects is n factorial, usually written as n!.

Do you get it?

Now, let's continue with permutation.

With that ABC objects, we only have three objects (A, B and C) which we can get easily, some question will be very tough to get e.g in how many ways 5 passengers can sit in a compartment having 16 vacant seats.

It will be tedious to solve when we want to solve a question like that, this is when we will apply this formula below to make work easy.

Where

n= the number of distinct/ different objects

r= the number of objects taken out of n

Using this formula to solve the question example 1 above

In this case n=3 and r=3

As you can see, we get the same answer using the two approaches.

Now ,let's try some examples.

In how many ways can 4 passengers can sit in a compartment having 10 vacant seat.

So the number of ways in which 4 passengers be seated in compartment having 10 vacant seats is 5040 ways.

Is it clear?

If yes do this,

Find the number of ways in which 8 students can be arrange from 10 students?

What is your answer?

The answer is 1,814,400

I believed you got it right

Now let's go ahead.

There are some permutation questions which won't require using the above formula. These are WORD and CIRCULAR PERMUTATIONS

WORD/LETTER PERMUTATION

The formula used in WORD/LETTER PERMUTATION is below.

Example -- Find the number of ways you can arrange the letter in the word word STUDENTS.

Firstly, by counting the letters in STUDENTS one another, the total is 8.

That means n=8
Secondly, we need to write the alphabet and count it one another.

S= 2( you can see that it appear twice)
T=2
U=1
D=1
E=1
N=1

By substituting that into formula

So, the number of ways of arranging STUDENTS is 10,080 ways

Now, try this,

Find the number of ways to arrange the word VACUUM?

What is your answer?

The answer is 360 ways

CIRCULAR PERMUTATION

This is the number of ways of arranging n different objects in a circle and the formulae use is simply (n-1)!

Example:- A family of 5 are to seat around a circular table for dinner. In how many ways can the family be seated.

The formula is (n - 1)!
n=5

So we have (5 -1)! = 4! = 4 * 3 * 2*1 = 24 ways

The number of ways for them to be seated in a circular table is 24 ways

If you have any questions, please drop them in the comment section.

I will explain why 0!=1 in my next post

Pictures : All picture used in this post was created by me using math editor app and I drawn one by myself on paper.

References

[1] John Bird Engineering mathematics- Fifth edition.
[2] Wolfram - Permutation
[3] Wikipedia - Permutation