Restricted permutation is the arrangement of elements made under certain restrictions. In restricted permutations certain elements are always either included or excluded. In this article we will learn about, permutation definition, combination definition and restricted permutation in detail.

Permutation Definition

Permutation is often referred to as the act of arranging all the items of a set into some particular sequence or order. If the set is already ordered, then the corresponding rearrangement of its elements is known as the process of permuting. Permutations occur most often arise when different orderings on certain finite sets take place.

Permutations is represented by the following formula, 

ⁿP_r = (n!) / (n-r)!

Combination Definition

Combination is a way of extracting and selecting the items from a set of items, in such a way that the order of selection does not matter in this case. It is equivalent to the count of the number of combinations of the given set of observations. It is basically equivalent to the combination of n things taken k at a time without any repetition. In order to represent the combinations where repetition is allowed, the terms k-selection or k-combination with repetition are often used.

The combinations are represented by the following formula, 

ⁿC_r = (n!)/r!(n-r)!

Restricted Permutations

The permutation is a way of filtering and selecting a set of objects, where the arrangement of objects does matter. However, the arrangement of objects may be done by imposing certain restrictions in the order of selection. For instance, the order of arrangement of articles, such that an article is always included or excluded from the set of given objects. Imposing the restrictions implies that not all the objects from the given set need to be ordered.

There are different types of common restrictions that may be imposed on the permutation:

• Inclusion of a set of objects
• Exclusion of a set of objects
• Certain objects that always occur together
• Certain objects that stay apart

Common types of restricted permutations are:

• Formation of numbers with digits with some digits at fixed positions.
• Word building with some letters with a fixed position.
• Vowels or consonants in the set of alphabets occur together.
• A set of objects always occurring together
• A set of objects that never occur together
• Restrictions for circular permutations
• Choice of dress to wear from a set of dresses
• Order of eating
• Combinations of the colors to make

Formula of Restricted Permutations

• Number of permutations of ‘n’ things taking ‘r’ at a time, corresponding to the case where a particular thing always occurs

r × ^n-1P_r-1

• Number of permutations of ‘n’ things taking ‘r’ at a time, corresponding to the case where a particular thing never occurred

^n-1P_r

Sample Questions

Question 1. Find out how many 4 digits numbers without any repetition can be made using 1, 2, 3, 4, 5, 6, 7 if 4 will always be there in the number?

Solution:

Here to find 4 digits number without any repetition can be made using 1, 2, 3, 4, 5, 6, 7 if 4 will always be there in the number,

We will use the formula for

Number of permutations of ‘n’ things taken ‘r’ at a time. In which a particular thing always occur

r × ^n-1P_r-1

Here,

• r = 4
• n = 7

Further putting values in the above formula

⇒ r × ^n-1P_r-1

⇒ 4 × ^7-1P_4-1 

⇒ 4 × ⁶P₃

⇒ 4 × 6!/3!

⇒ 4 × (6 × 5 × 4 × 3!)/3!

⇒ 480

Therefore,

480 numbers can be made

Question 2. How many 5 digit numbers can be formed by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So that 2 is always there in the number?

Solution:

Here to find 5 digit numbers can be formed by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So that 2 is always there in the number,

We will use the formula for

Number of permutations of ‘n’ things taken ‘r’ at a time. In which a particular thing always occur

r × ^n-1P_r-1

Here,

• r = 5
• n = 10

Further putting values in the above formula

⇒ r × ^n-1P_r-1

⇒ 5 × ^10-1P_5-1 

⇒ 5 × ⁹P₄

⇒ 5 × 9!/(9-4)!

⇒ 5 × (9 × 8 × 7 × 6 × 5!)/5!

⇒ 15120

Therefore,

15120 numbers can be made

Question 3. How many different three-letter words can be made by 5 vowels if ‘a’ is never included?

Solution:

Here to find different three-letter words can be made by 5 vowels if ‘a’ is never included,

We will use the formula for

Number of permutations of ‘n’ things taken ‘r’ at a time. In which a particular thing never occurred

^n-1P_r

Here,

• r = 3
• n = 5

Further putting values in the above formula

⇒ ^n-1P_r

⇒ ^5-1P₃ 

⇒ ⁴P₃

⇒ 4!/(4 – 3)!

⇒ 4 × 3 × 2

⇒ 24

Therefore,

24 words can be made

Question 4. How many four-digit numbers without any repetition can be made by using 1, 2, 3, 4, 5, 6, 7 if 4 will never be included?

Solution:

Here to find four-digit numbers without any repetition can be made by using 1, 2, 3, 4, 5, 6, 7 if 4 will never be included,

We will use the formula for

Number of permutations of ‘n’ things taken ‘r’ at a time. In which a particular thing never occurred

^n-1P_r

Here,

• r = 4
• n = 7

Further putting values in the above formula

⇒ ^n-1P_r

⇒ ^7-1P₄ 

⇒ ⁶P₄

⇒ 6!/(6 – 4)!

⇒ 6 × 5 × 4 × 3 × 2!/2!

⇒ 360

Therefore,

360 numbers can be made