various types of permutations
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(1) Permutation of n different objects:
If n is a positive integer and r is a whole number, such that r < n, then P(n, r) represents the number of all possible arrangements or permutations of n distinct objects taken r at a time. It can also be represented as
nPr.
P(n, r) = n(n-1)(n-2)(n-3)……..upto r factors
⇒ P(n, r) = n(n-1)(n-2)(n-3)……..(n – r +1)
⇒P(n,r)=n!(n−r)!
(2) Permutation when repetition is not allowed:
When the number of object is “n,” and we have “r” to be the selection of object, then
Choosing an object can be in n different ways (each time).
Thus the permutation of objects when repetition is allowed will be equal to,
n×n×n×……(rtimes)
which is given as nr
(3) Permutation of multi-sets:
Permutation of n different objects when p1objects among ‘n’ objects are similar, p2 objects of the second kind are similar, p3 objects of the third kind are similar ……… and so on, pk objects of the kth kind are similar and the remaining of all are of a different kind,
Thus it forms a multiset, where the permutation is given as:
n!p1!p2!p3…..pn!
If n is a positive integer and r is a whole number, such that r < n, then P(n, r) represents the number of all possible arrangements or permutations of n distinct objects taken r at a time. It can also be represented as
nPr.
P(n, r) = n(n-1)(n-2)(n-3)……..upto r factors
⇒ P(n, r) = n(n-1)(n-2)(n-3)……..(n – r +1)
⇒P(n,r)=n!(n−r)!
(2) Permutation when repetition is not allowed:
When the number of object is “n,” and we have “r” to be the selection of object, then
Choosing an object can be in n different ways (each time).
Thus the permutation of objects when repetition is allowed will be equal to,
n×n×n×……(rtimes)
which is given as nr
(3) Permutation of multi-sets:
Permutation of n different objects when p1objects among ‘n’ objects are similar, p2 objects of the second kind are similar, p3 objects of the third kind are similar ……… and so on, pk objects of the kth kind are similar and the remaining of all are of a different kind,
Thus it forms a multiset, where the permutation is given as:
n!p1!p2!p3…..pn!
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