Permutation Under Certain Conditions


 
 
Concept Explanation
 

Permutation Under Certain Conditions

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.

Permutation Under Certain Conditions

Permutations where certain conditions are applied fall in this category. The conditions may be in terms of not using all the items or, a particular item should occur in each and every permutation.

Number of permutations of n different things, taken r at a time, when a particular things is to be always included in each arrangement

 =r.^{n-1}P_{r-1}.

For Example: A team of 3 cars is to be formed from 8 cars in which one car is always included, then possible arrangement is

 3.^{8-1}P_{3-1}

=3times ^7P_2

=3times frac{7!}{(7-2)!}

=3times frac{7times6times5times4times3times2times1}{5times4times3times2times1}

=3times7times6 = 126

Number of permutations of n different things taken all at a time, when m specified things always come together is

m! times (n - m + 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

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Sample Questions
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Question : 1

What will be the total ways in which 4 letter words can be formed from the letters of word Delhi in which all vowels are always together ?

Right Option : B
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Question : 2

What will be the total ways in which 4 letter words can be formed from the letters of word MAHESH in which all vowels are always together?

Right Option : D
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Question : 3

All the five letters of the word TABLE  are arranged to form different words without repeating any letter in any word. The words so formed are, then arranged as in dictionary. What will be the position of the word TABEL in that sequence ?

Right Option : B
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