Permutation and their application

A mathematical technique that is used to arrange objects out of the total objects or persons. In permutation, the order of objects or the way they are arranged is of high importance. We can denote permutation by the formula nPr. Where r is the number of selected objects and n is the total number of objects. Let’s take an example to get a clear picture of permutation. 

How many 4 digits numbers can be formed by using the digits 2,3,4,5,6 using each digit only once.

We will get different numbers by arranging the given digits in different ways – 2345, 6543

So, total four digits number = 5P4 

n=5,r=4

=5!/(5-4)! = 5!/1! = 5*4*3*2*1 = 120. 

Classification of permutation 

Permutation can be classified into three categories- 

1. Permutation of n different objects when repetition is not allowed.
2. Permutation of n different objects when repetition is allowed. 

Formula to calculate permutation 

The formula of permutation when, repetition is not allowed – nPr=n!(n-r)!

The formula of permutation when, repetition is allowed – nPr= nr

Examples of permutation

Type 1- it is based on the fundamental principle of counting or we can say simple questions regarding elements of selection of objects of two given objects without any conditions.

1.In a race, 9 students participated, the number of ways in which the first three places can be taken.

Sol- As per the question, 3 students can take the first 3 seats. This suggests that it is the question of permutation.

The first 3 places can be taken as-

nPr = 9!/(9-3)! = 9!/6! = 9*8*7= 504 Ways

Type2- It is based on arrangement of objects taking more than one, together or separate according to condition

1.If 2 nP3 = n+1P3

nPr = n!/(n-r)!

2.n!/(n-3)! = (n+1)!/(n+1-3)

2.(n)!/(n-3)! = (n+1)(n)!/(n-2)(n-3)

2(n-2)=n+1

2n-4=n+1

2n-4=n+1

n=5.

Type 3- it is based on circular arrangements. 

1.The ways of arrangement of 4 boys and 4 girls around a circular table if no two girls may sit together.

Sol- No. of ways in which boys can arranged = (4-1)! = 3*2*1 = 6

No.of ways in which girls can be arranged = 4P4  = 4! = 4*3*2*1 = 24.

Therefore, total number of ways in which in which 4 boys and 4 girls can be arranged in a circle = 24*6 = 144 ways.

Conclusion 

In mathematics, a permutation of a set is a loose grouping of its members into a sequence or linear order, or a rearrangement of its elements if the set is already sorted. The act or process of shifting the linear order of an ordered set is also known as “permutation.” The formula of permutation when, repetition is not allowed – nPr=n!(n-r)!

The formula of permutation when, repetition is allowed – nPr= nr

Some questions based on permutation

Q1. How many different digit numbers can be formed by using the digits 1,2,3,4,5,6 using one or more digits at a time. 

Solution- we can make one, two, three, four, five, and six-digit numbers.

1 digit number = 6P1 = 6!/5! = 6.

2 digit number = 6P2 = 6!/4! = 6*5=30.

1 digit number = 6P3 = 6!/3! = 6*5*4=120.

1 digit number = 6P4 = 6!/2! = 6*5*4*3=360.

1 digit number = 6P5 = 6!/1! = 6*5*4*3*2*1=720.

1 digit number = 6P6= 6!/0! = 6*5*4*3*2*1=720.

Total number of ways= 6+30+120+360+720+720=1956.

Q2. In how many different ways the word “DETAIL” can be arranged in such a way that the vowels occupy only the odd positions.

Solution- total number of letters in word ‘DETAIL’ = 6

Vowels in a letter = 3

Consonants in a letter = 3

We can place 3 vowels at any of the 3 places out of 4 and marked 1,2,3 

Number of ways of arranging the vowels = 3P3 = 3! = 3*2=6

Number of ways in which 3 consonants can be arranged = 3P3 = 3! = 3*2=6

Total number of ways = (6*6)=36.

Some questions for practice

Q1. Find the number of different words that can be formed from the letters of the word ‘PRACTICE’ so that no vowels are together.

Q2. There are five people named P1,P2,P3,P4,P5.  Out of 5 persons three persons are to be arranged in a line such that in each arrangement P1 must occur whereas P2 and p53 do not occur. Find the number of such possible arrangements.