Permutation Of Things, Definition, Formula, Types And Examples, Important Topics For JEE Maths 2024

Permutation Of Things : Plutarch made the first use of permutation by discovering the number of different syllables possible in Greek language. After that Al-khali an Arab Mathematician wrote the book of Cryptographic messages which contains use of permutation and combination. Permutation is defined as arrangement of objects.

it represents all possible orders with respect to position of objects for example permutation of , , in a line could be represented as shown below , , , , , here has covered all three positions similarly and now for it has covered first two position in last two positions in first and third position in also its arrangement in its  position is covered as for  first two last two first and last as , this discussion denotes when objects are being permuted all possible arrangements with respect to objects and there position is covered.

Formula for arrangement of things when all things are not different is represented as it has total n objects with p, q, r as identical objects, permutation of word “MISSISSIPPI” could be found by the formula as where we have 4 S, 4 I ,2 P. If there are n objects in which p, q, r are alike than arrangements of m objects out of the n objects could be done by making individual cases which may have different and alike objects accordingly.

Permutation Introduction

Permutation of things when all things are not different means at least one object must repeat for these kinds of arrangement all identical objects permutation must be counted once only for example arrangement of AAC would be AAC, ACA, CAA as permutation of AA in its position would be counted once only. Permutation of alike objects is represented as .if selection of m objects to be done from n objects having p, q, r alike objects than this could be done by making cases such as, arrangement of 2 alphabets from “POTATO” could be done as

Permutation Case 1

When all are different, we have 2’O’, 2T, 1’P’, 1’A’ So 2 different alphabets could be selected in ways where is and their arrangement is Total is

Permutation Case 2

When both are same, we have 2’O’, 2’T’, 1’P’, 1’A’ So 2 same alphabets could be selected in ways from (2’O’, 2’T’) and their arrangement is as both are identical.

Total arrangements are 12+2=14

Let’s see some examples

Example 1 : How many permutations of letters of the word “BANANA” are there?

Sol. BANANA has 3A,2N,1B possible arrangements of the word would be =60

Example 2: How many words can be formed using U thrice, T twice, P twice?

Sol. We have total 7 letters which includes 3U, 2T, 2P now possible arrangement would be

Example 3: How many words can be formed from the word “INDEPENDENCE” if all letters are permuted?

Sol. “INDEPENDENCE” has total 12 letters which includes 4E, 3N, 2D, 1P, 1I,1C

Now possible total arrangements of “INDEPENDENCE” would be

Example 4: How many words can be formed from the word “CONSTANTINOPLE” if all letters are permuted?

Sol. “CONSTANTINOPLE” has total 14 letters which includes 3N, 2T, 2O, 1C, 1I, 1E, 1S, 1A, 1P, 1L

Now possible total arrangements of “CONSTANTINOPLE” would be

Rapid Questions (Based On Permutation Of Alike Objects).

1. How many ways can letter of the word “INTERMEDIATE” be arranged?

2. How many words can be formed from the word “HARYANA” if all letters are permuted?

Complex Illustrations based on permutation of alike objects

Illustration

Q.1: How many different messages can be send using 4’0’ and 3 ‘1’?

Sol. Formation of message with 4’0’ and 3 ‘1’ is equal to arrangement of 4’0’ and 3 ‘1’ in a row as

Q.2: How many 3 letter words can be formed from the word “GOOGLE”?

Sol. We have 2’G’, 2’O’, 1L, 1E

Case 1: When all are distinct, selection is and arrangement is 3! Total

Case 2: When 2 same and one distinct, selection is and arrangement is Total

Case 3: When 3 same, we have maximum number of identical alphabets as 2 (2’G’, 2’O’) so selection of 3 same alphabets is zero.

Rapid Questions (Based On Permutation Of Alike Objects)

1. How many 3-letter word can be formed from “SCHOOL”?

2. Find Possible arrangements of word “SENSITIVE” when 3 letters are being selected?