**Permutation**

Hello Readers

Today’s topic is Permutation.

Let’s start discussing the topic Permutation with basic concepts.

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**Factorial:**

Factorial denoted by n! Or |__n __

Where n! = n. (n-1) (n-2) ………………… 3.2.1

(i) 0! = 1

(ii) 1! = 1

(iii) 5! = 5.4.3.2.1 = 120

(iv) 6! = 6.5.4.3.2.1 = 720

**Permutation: (Arrangement)**

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The different arrangements of a given number or things by taking some or all at a time, are called permutations.

All permutations (or arrangements) made with the letters of ‘abc’, taking two at a time are:

(ab, ba, ac, ca, bc, cb)

All permutation made with the letters a, b, c taking all at a time is:

(abc, acb, bac, bca, cab,cba)

**Number of Permutations:**

Number of all permutation of n things, taken r at a time, is given by

^{ }

Example –

Let’s discuss some examples based on the above discussed concept –

**Example 1**

In how many different ways can the letters of the word HOME be arranged?

**Solution – **

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n = 4, r = 4

By formula-

**Trick:**

^{n}P_{n }= n!

4! = 4 × 3 × 2 × 1 = 24

**Example 2**

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In how many ways can the letters of word ‘COFFEE’ arranged?

**Solution- **

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Here ‘F’and ‘E’ repeats for two times therefore –

**Example – 3**

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In how many different ways can the word ‘HOME’ be arranged so as vowels always come together?

**Solution – **

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In this example, there are two vowels (O, E)

HOME = H, M, (O, E)

Now, we will consider O and E as a single unit and then count all the units i.e. 3 units.

3! × 2! = 3 × 2 × 1 × 2 × 1 = 12 ways.

**Example – 4**

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In how many different ways can the word ‘COFFEE’ be arranged so as vowel always come together?

**Solution –**

In this word, words repeat as –

F – Two times

E – Two times

And there are three vowels O, E , E

Now, consider O, E , E as a single unit

CFF, OEE

**Example 5**

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In how many different ways can the word ‘HOME’ be arranged so as, the vowels never come together?

**Solution – **

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Arrangements for never = Total arrangement (See ex 1) – Arrangement for always (See ex 3)

= 4! – 3! × 2!

= 24 -12 = 12

**Example 6**

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In how many different ways can the word ‘COFFEE’ arranged so as, vowels never come together?

**Solution – **

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Ways (never) = Ways (Total) – Ways (Always)

(See ex 2) (See ex 4)

= 180 – 36

= 144.

With this we will finish this topic here.

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