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Permutation and Combination of Alike Objects for JEE

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Last updated date: 22nd Feb 2024
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A Brief Idea about Permutation and Combination

Permutation and combination are key ideas in mathematics that we employ to choose data or objects from a given set. Permutations and combinations are collections of things that can be used together to form a group. The mathematical ideas of permutation and combination were initially introduced by Blaise Pascal and combination- one of the essential topics in mathematics, is used in science, engineering, and research. The distinction is that  permutation is the order in which the data is placed is known,  whereas combination is when a set of information is selected from a specific group.

 

Permutation and Combination

Permutation: 

The term 'arrangement' can be seen in the definition of permutation. Permutation refers to the simultaneous arrangement of some or all of the same or different things. In this context, the meaning of arranging is choosing and managing.

 

Combination: 

Combination refers to a set of elements that may be similar or dissimilar and are related to choosing one or more items. A combination can also refer to the collection, selection, or committee of words.

 

Permutation and Combination Formulas

  • For Permutation

'r'(n !){(n-p)!


Example- All conceivable combinations using the letters a, b, and c.

By taking two at a time are ab, ac, ba, bc, ca, cb.
By taking all three at a time are abc, acb, bac, bca, cab, cba.

 

  • For Combination

There are various options for selecting 'r' items from 'n' items. 

${ }^{n}C_{r}=\dfrac{n !}{r !(n-r) !}$ 

Example- All the potential combinations using the letters a, b, and c. 

When three letters out of three are to be chosen, abc is the sole option.
When selecting two out of three letters, the possible combinations are ab, bc, ab.

 

Some Other Formulas

  • If n is even nCr is greatest for $r=\dfrac{n}{2}$

  • ${ r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$
  • ${ }^{n} C_{r}= { }^{n} C_{n-r}$

 

Difference Between Permutation and Combination

When things are of a distinct type, permutation refers to the various conceivable arrangements. To understand how to use permutation and combination correctly, you must first understand the difference. We only care about the set of items that make up a particular group in combinatorics; conjunction refers to the number of smaller groups or locations that can be constructed from the constituents of a more extensive set. Refer to the table below to better understand the difference between permutation and combination.

 

Sr. No.

Combination

Permutation

1.

Combinations are used to determine how many different groups can be formed.

When an order/sequence of arrangement is required, permutations are used.

2.

Combination of two items from a set of three X, Y, and Z are written as XY, YZ, and ZX.

XY, YX, YZ, ZY, XZ, ZX is the permutation of two items from three given things X, Y, Z.

3.

Combinations are used to describe items that are similar in nature.

Permutations are used for a variety of things.

4.

There are various options for selecting 'r' items from 'n' items.${ }$

'r' things taken from 'n' things can be arranged in a variety of ways ${ }^{n}P_{r} = \dfrac {n !}{(n-p) !}$


A permutation's response is always greater than a combination's answer. The permutation value is always more significant than the mix value for given values of n and r. Different alternative arrangements are counted in permutations, whereas only different subgroups are counted in combinations.


Permutation and Combination Examples

Example 1: Determine the number of distinct ways the word 'ASSASSINATION' can be spelt.

Solution:  

We've offered 13 alphabets in this example.


Alphabets

Times

A

Occurs 3 

S

Occurs 4

I

Occurs 2

N

Occurs 2

T and O

Occurs 1


As a result, the number of possible permutations is enormous $=\dfrac{13 !}{3 ! 4 ! 2 ! 2 !}$


Example 2: Determine the number of words made up of 5 letters from the letters in the word "INDEPENDENCE."

Solution: 

We've offered 12 alphabets in this example

 

Alphabets

Times

E

4

D

2

P, I, C

1

 

Case 1 - When EEEDC (2 different and 3 alike)

Number of selections $={ }^{2} C_{1} \times{ }^{5} C_{2!}{3!}$

Thus, number of permutation $=20\times\dfrac{5!}{3!}=400$

 

Case 2 - When EEEEP (4 alike and 1 different)

We only have one option for four people who are the same, namely E, and five alternatives for choosing one person who is different. 

Number of selection $=1\times { }^{5} C_{1}=5$ 

Arrangement $=\dfrac{5!}{4!}$

Thus, 

 

Case 3 - When EENDI (2 alike and 3 different)

Number of selection $= {}^{3} C_{1}\times{ }^{5} C_{3}=30$

Arrangement $= {}^{2} C_{1}\times{ }^{5} C_{1}=4$ 

Arrangement $= \dfrac{5!}{3!2!}$

Thus, the number of permutations $= 4\times\dfrac{5!}{3!2!}=40$

 

Case 4 - When EDIPC (All five different)

Number of selection $= { }^{6} C_{5}=6$

Arrangement END (2 alike, 2 other alike and 1 different)

Number of selection $= { }^{3} C_{2}\times4=12$

Arrangement $= \dfrac{5!}{2!2!}$

Thus, the number of permutations $= 12\times\dfrac{5!}{2!2!}=360$

As a consequence, sum all of the outcomes from the various situations together is:

$= (400+ 25 + 1800 + 40 + 720 + 360)$

$= 3345$

 

Conclusion

The significant difference between permutation and combination in  mathematical notions is point order, placement and position, essential in permutation but not in combinations. Combination arranging objects, persons, numbers, letters, colors, and so on.

 

A combination is an unordered set or pair of values within the defined criteria. A single combination can provide a large number of permutations. A single permutation, on the other hand, yields only one combination. Following the above discussion, it is evident that permutation and combination are two different words used in mathematics, statistics, research, and everyday life.

Competitive Exams after 12th Science

FAQs on Permutation and Combination of Alike Objects for JEE

1. What is the difference between permutation and combination?

The difference between permutation and combination can be explained as: 

  • Permutation refers to the various ways of arranging a group of objects in sequential order while combination is one of the multiple strategies for choosing things from a significant gathering of items without respect for their request.

  • Arrangement of r items out of total n items by using combination selection of r items out of total n items can be explained as ${ }^{n} C_{r}=\dfrac{n !}{r!(n-r)!}$

  • Permutation is used for a variety of items while combination is used to select out of similar items. 

2. In mathematics, what are the terms ${}^nC_r$ and ${}^P_r$?

Terms ${}^nC_{r}$ and ${ }^{n} P_{r}$ are used to denote permutations and combinations. A permutation  is the process of putting the components of a group or set in a specific order and $\left({ }^{n} P_{r}\right)$ represents the arrangement of r items out of total ‘n’ items while a e a Combination is the process of selecting elements from a group or set in which the order of the ${ }^{n} P_{r}=\dfrac{n !}{(n-rp) !}$ and ${ }^{n} C_{r}=\dfrac{n !}{r!(n-r)!}$ . Here, in both terms, the value of n should be greater than r

3. What is the importance of Permutation and combination? What is the number of questions asked in JEE-Mains from Permutation and combination ?

Permutation and combination is an important topic for the JEE exam. In the previous year's paper, at least one question was from Permutation and combination. This topic is also important for your board exams as numerical and multiple-choice questions based on to do higher studies in statistics, then this topic will strengthen your basics and will help in a better understanding of higher concepts.