Permutation of Objects Not All Distinct


 
 
Concept Explanation
 

Permutation Of Objects Not All Distinct

Here we are considering  the permutations of a given number of objects when objects are not all different. For example, the number of arrangement of the letter of the word MISSISSIPPI, or the number of six digit number formed by using the digits 1, 1, 2, 3, 3, 4 etc. the permutations can be calculated as.

The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p + q = n, is

                    frac{n!}{p! q!}

Case I:

The number of permutations of n things, of which P_1 are alike of one kind; P_2 are alike of second kind; P_3 are alike of third;...; P_r are alike of r th kind such that

P_1+P_2+...+P_r=n is

=frac{n!}{P_1  ! times P_2  ! times P_3 !times ...P_r !}

Case II:

The number of permutations of n things, of which p are alike of one kind, q are alike of second kind and remaining all are distinct, is

=frac{n!}{p! q!}

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Sample Questions
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Question : 1

Find the number of arrangements of the letters of the word  INDEPENDENCE. In how many of these arrangements

(i) do the words start with P

(ii) do all the vowels always come together

Right Option : C
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Explanation
 
 


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