Introduction

Permutations are a method to determine the number of all possible arrangements of a set of elements. The arrangement of elements is significant, and the arrangement of elements is made by applying certain restrictions. 

Restricted permutations are permutations where certain elements are always either included or excluded. In addition to this, it also refers to the permutations that can either always occur together or always stay apart.

Types of restrictions that may be applied:

  • Certain elements that are always included
  • Certain elements that are permanently excluded
  • Certain elements that consistently occur together
  • Certain elements that always stay apart

NOTE:- nPr =  n! / (n – r)!

Examples of permutations with formulae

  1. Number of permutations of ‘n’ elements, taking ‘r’ at a time, when a particular element is always included in each arrangement: r x n-1Pr-1

Example:

 Find out how many three digits numbers without any repetition can be made using 1, 2, 3, 4, 5 if one will always be there in the number.

Solution:  

Here, we will use the formula: r x n-1Pr-1

It is given that, r = 3

                        n = 5

Now by putting the values of r and n in the formula r x n-1Pr-1

We get,

  = 3 x 5-1P3-1

 = 3 x 4P2

 = 3 x (4! / (4-2)!)

 = 3 x [(4 x 3 x 2!) / 2!]

 = 3 x (4 x 3)

 = 3 x 12

 = 36

Therefore, 36 three digits numbers can be made where one will always be included in the 

number.

  1. Number of permutations of ‘n’ things, taken ‘r’ at a time when a particular element is permanently excluded in each arrangement: n-1Pr

Example:

 Find out how many three digits numbers without any repetition can be made using 1, 2, 3, 4, 5 if 1 will never be there in the number.

Solution:  

Here, we will use the formula: n-1Pr-1

It is given that r = 3

                        n = 5

Now, by putting the values of r and n in the formula n-1 P r

We get:

= n-1Pr

=  5-1P3

= 4P3

= (4! / (4-2)!)

= [(4 x 3 x 2!) / 2!]

= (4 x 3)

= 12

Therefore, 12 three-digit numbers can be made wherein 1 will never be there in the number.

Conclusion:

Here we have learnt how to deal with restricted permutations, wherein we have to permutate according to the conditions given. Such problems can be solved by using basic concepts of permutations and combinations, along with some logical approaches.