Linear Permutation


 
 
Concept Explanation
 

Linear Permutation

Permutation

All the possible ways the things can be arranged by using some or all the things are called permutation. Thus each of the arrangements which can be made by taking some or all number of things is called a permutation.

The various ways in which the three things can be arranged by taking two at a time are known as the permutation of three things takes two at a time.

The permutations which can be made by taking the letters a, b and c by taking two at a time are 6 i.e., ab, bc, ac, ba, cb and ca. Each of these presenting a different arrangement of two letters. These six arrangements are called  permutations of three things takes two at a time.

Types of Permutation

Permutation is generally of two types

1. Linear permutation     

2. Circular permutation

1. Linear Permutation

If the things are arranged in a row/line, then a permutation is called linear permutation. It is simply written as a permutation.

Number of permutations of n dissimilar things taken r at a time is ^nP_r.

                 ^nP_r =n(n-1)(n-2)...(n-r+1)=frac{n!}{(n-r)!}

Where, n! is the product of the first n natural numbers and called n - factorial or  factorial n denoted by n!.

e.g.,            5! = 5 X 4 X 3 X 3 X 2 X 1 = 120

When n is a negative integer or a fraction, where n! is not defined. Thus, n! is defined only for positive integers.

According to the above definition, 0! makes no sense.

However, we define 0! = 1

                             n! = n ( n-1)!

For example, The total ways in which the letters of word 'DISCOVER' can be arranged in which all vowels are always together is

Here the total letters are = 8

No of vowels = 3

So in the formula m= 3 and n =8

= 3! times ( 8 - 3 + 1)!

= 3! times 6!       (because there are 3 vowels)

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Sample Questions
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Question : 1

Among the 36 teachers in a school , one principal and one vice principal is to be appointed . In how many ways can this be done ?

Right Option : B
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Question : 2

If  nP= 360 , find r.

Right Option : D
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Question : 3

In how many ways can 6 persons stand in a queue?

Right Option : A
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