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When it comes to finding permutations of a certain number of items, many students feel overwhelmed. This comprehensive guide breaks down the process into easy-to-follow steps, making it easy for anyone to find all possible permutations. Whether you’re studying for an exam or just want to learn more about permutations, this guide is for you!
What is Permutation?
The number of different ways a set of items can be arranged is called a permutation. It is usually denoted as P(n, r) or nPr. For example, the permutation of {apple, orange, grape} is denoted as P(apple, orange, grape).nPr is the total of permutations of n distinguishable things taken r at a time. Let’s look at some concepts.
Permutation of N different Number
To find the permutation of N distinct numbers, we can use the following formula:
n! / (n – r)!
where,
n = number of distinct numbers
r = number of things to be selected
For example, let’s say we have a set of five distinct numbers: {4, 12, 34, 56, 78}. To find the permutation of three distinct numbers taken from this set, we can use the formula above.
n! / (n – r)!
= 5!(5-3)!
= (5x4x3x2x1)/(2×1)
= 120/2
=60
Therefore, the permutation of three distinct numbers taken from the set {4, 12, 34, 56, 78} is 60.
Permutation of N different Numbers when repetition is allowed:
To find the permutation of N distinct numbers when repetition is allowed, we can use the following formula:
n x n x n x n x n x ….. (r times) = nr
where,
n = number of distinct numbers
r = number of things to be selected
For example, let’s say we have a set of five distinct numbers: {11, 22, 33, 44, 55}. To find the permutation of three distinct numbers taken from this set when repetition is allowed, we can use the formula above.
nr
=5x5x5
= 125
Therefore, the permutation of three distinct numbers taken from the set {11, 22, 33, 44, 55} when repetition is allowed is 125.
Permutation of N things when not all are different:
The number of permutations of N things taken all together, when x of the things are alike of one kind, y of them alike of another kind, z of them alike of a third kind, and the remaining all different is n! / [x! y! z!]
For Example :
There are 11 letters in the word “MATHEMATICS”. How many different arrangements can be made from these 11 letters if all the vowels must be together?
Here,
MATHEMATICS=11 letters
As per the condition, Vowels must stay together,
AEAI=4 Vowels
MTHMTCS=7 Consonants
As per the Condition, two groups are formed i.e. (MTHMTCS) & (AEAI)
In the Consonants’ group, there are 2 Ms and 2 Ts
In Vowels’ group 2 A’s are present
Words That can be formed by the internal arrangement amongst Consonants and a Set of Vowels are
8!/((2! x 2!) = 10080
Words That can be formed by the internal arrangement amongst Vowels are
4!/(2!) = 12
And the total number of different arrangements that can be made from these 11 letters if all the vowels must be together are
10080 X 12 = 120960.
Solved Examples
Example: Find the number of ways in which four distinct things can be arranged in a row.
Answer: Total number of ways = n! = 24.
Example: Find the number of arrangements that can be made out of the following word:
(i) BANANA
(ii) ASSASSINATION
Answer: (i) The word ‘BANANA’ has 6 letters, out of which 3 As are alike and 2 Ns are alike.
∴ Required number of arrangements = n! / r!
= 6! / 3! x 2!
= 6 x 5 x 4 x 3 x 2 x 1 / 3 x 2 x 2 x 1
= 720/12
= 60 Arrangements
(ii) The word ‘ASSASSINATION’ has 13 letters, out of which 3 As are alike, 4 Ss
are alike, 2 Is are alike and 2 Ns are alike.
∴ Required number of arrangements = n! / r!
=13! / 3! x 4! x 2! x 2!
= 13x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 / 3 x 2 x 4 x 3 x 2 x 2 x 2
= 6227020800 / 576
= 10810800.
Conclusion
Permutations are a type of combination, which is the process of selecting r objects from n given objects. In this article, we have provided a comprehensive guide on how to find permutations of n distinct things. We hope that you found this information helpful and that it will help you in your future mathematical endeavours. If you have any questions or feedback about this article, please don’t hesitate to reach out to us.
