Number Of Permutation Of N Different Things :
First documented use of permutation was done by Plutarch who discovered number of syllable possible in Greek language, later on 1677 Fabian Stedman made use of it for permutation of bells, Stedman elaborated it further in permutation of letters and real-life objects. Joseph Louis Lagrange used it in study of roots for polynomials while using various possibilities of roots.
Permutation denotes number of arrangement means if a child has to arrange 3 toys in a row named as
than possible arrangements of these toys would be
this could be shown as 3!. Now here as we could see for each arrangement there is no repetition now if repetition occurred means there is sufficient quantity of each variety of toys than possible ways would be
Now if the child has to arrange 2 toys only out of three while every variety has sufficient quantity than number of ways would be
this could be shown as
means
and
means in case of
as
, in case of
as
, in case of
as
.
Now this concept could be generalized for n objects as, Arrangements of r objects out of n objects in r places when any object can be repeated any number of times would be
, Let’s explore it with examples in next section
Number Of Permutation Of N Different Things Introduction
We can categories permutation with repetition in various parts such as
-
∙
Permutation with repetition of n objects
-
∙
Permutation with repetition of r objects out of n objects in r places
.
-
∙
Permutation with repetition of n objects out of n in r places
.
Let’s explore above parts with examples
Permutation examples based on when repetition is allowed
Number Of Permutation Of N Different Things Example 1
Q. Find total number of ways of making 2-digit number out of 1,2,3,4,5 repetition is allowed?
Sol.
Here we have to form a 2-digit number out of given numbers, so it lies to third category as Permutation with repetition of n objects out of n in r places here n is 5 and r is 2 so total number of ways would be
this could be shown as 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55
Number Of Permutation Of N Different Things Example 2
Q.
Find total number of words can be formed using letters from “TRIANGLE” when repetition is allowed?
Sol.
TRIANGLE consist of 8 letters and this arrangement would lie in first category as Permutation with repetition of n objects here n is 8 so answer would be
.
Number Of Permutation Of N Different Things Example 3
Q.
Find total Number of ways in which 3 letters could be posted in 2 post offices?
Sol.
Here we have 3 letters out of which 3 has to be posted in 2 post offices So here answer would be
and this could be shown as let letters are
and post office as
than arrangements would be as
,
,
,
,
,
,
,
Number Of Permutation Of N Different Things Rapid Questions When Repetition Is Allowed) -
1.
Find total Number of ways in which 5 letters could be posted in 4 post offices?
2.
Find total number of 8 digits can formed out of 1,2,3,4,5,6,7?
Complex permutations based on when repetition is allowed
Illustrations
Q.1.
How many ways can 6 pictures be hung from 4 picture nails on the wall a nail can accommodate any number of pictures?
Sol.
Here we have total 6 objects(pictures) had to be arranged in 4 nails, total number of ways would be
Q.
2.
How many ways 2 rings can be put in 5 fingers while rings can be adjusted in any finger?
Sol.
Here we have 2 rings which had to be arranged in 5 fingers this could be done in
ways and shown as let rings are
and fingers are
than number of arrangements would be as
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Complex Permutations Rapid Questions(Based On When Repetition Is Allowed)
1.
How many ways can 7 pictures be hung from 6 picture nails on the wall a nail can accommodate any number of pictures?
2.
How many ways 4 rings can be put in 5 fingers while rings can be adjusted in any finger?
