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Permutations
21.10.2013 14:57

Define Permutation (Read more): - Permutation means the arrangement. Each of the arrangements which can be made by taking some or all of a number of things is called a permutation.  For example, suppose there are three quantities a, b and c. the different orders of arrangements of these three will be six. The permutation of the three quantities can be given as abc, bca, cba, acb, bac and cba. Let four persons enter a train compartment where there are six seats. How these four persons can be arranged on the six seats, this science or calculation is known as the permutation or the arrangement. The required arrangement is 6p4  which is equal to thirty.

Let us to find the number of permutations of n things taken all together, when the things are not all different.  Let the   n  things be represented by  n  letters and suppose   P  of  them to be a’s,  q of them to  be b’s,  r   of them to be c’s, and  the rest  to be unlike. Let N represent the required number of permutations. Then, if in any one of these permutations, the  P  letters,  a  were  changed into P unlike letters different from any of the rest,  then without altering the position of the remaining letter, from this single permutation alone we could form p different permutations. Hence if this change were made in each of N permutations the whole number of permutation would be N fact P. Similarly, if in each of these new N fact P permutations, the q  letters, b, were changed in to q unlike letters, different from any of the rest, then by rearranging q   letters alone amongst themselves, in all possible ways, each would produce factq different permutations.

Hence the total number of permutations when these changes are made) would be N. fact p. fact q , Likewise,   in each of these N.fact pfactq  permutations, again ,  r  letters C  were changed into  r  unlike letters, different from any of the rest, the whole number of permutations would be N fact p fact q fact r. Thus, when all the letter are different from one another the total number of permutations

N   =        fact n / fact P fact q fact r

Permutation Definition (Read more): - The number of permutations of n same as the number ways in which r places can be filled up by n different things. As many one of the n things can be put on the first place. It can be filled up in b ways, the second place can be filled up in (n – 1) ways. Since each way of filling up the first place can be associated with each way of filling up the second place.

Thus, the first two places can be filled up in (n- 1) ways, the third place can be filled up by any one of the remaining (n- 2) different ways. Hence, the number of different ways in which the first three places can be filled up in n (n- 1) (n – 2) ways.     Proceeding in this way, we find that total number of ways of filling up any number of places in equal to the product of same number of factors.

Integration Examples

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