Permutation And Combination For Competitive Exams

Competitive exams have become an inseparable component for admissions at the universities and for recruitment purposes. The quest to recruit competent candidates for various job profiles, degree programs and companies hugely rely on such competitive exams. Although the pattern for an entrance exam may vary depending on the purpose, the core components remain the same. Aimed at testing the subjective knowledge and aptitude of a candidate, the competitive exams have various sections like Analytical Reasoning, Comprehension, Logical Reasoning, etc. Each section carries questions on diverse topics like Sentence Correction, Verbal Ability, Problems on Ages, etc, to assess the competencies and mathematical reasoning skills of a candidate. Just like Ratio and Proportion problems or Percentage questions, Permutation and Combination also forms an important portion in competitive exams. Thus, in this blog, we will discover the various aspects related to this topic. 

What is Permutation and Combination?

Permutation and Combination deals with looking for various ways in which characters from a given set can be used to form subsets without replacements. The permutation involves the orderly arrangement of elements of a set. While in combination, we look for the number of ways a given set of characters can be arranged without considering their order. Students often are confused between the two and use the words interchangeably. Below is an example to make it easy to understand. 

For example, your phone’s password is 4321, and if you enter 1234, it will not unlock despite the numbers being the same. What your phone recognizes is the order of the numbers. There are many combinations possible for the given set of numbers but your phone accepts only a specific permutation. 

In a given set of five letters, A, B, C, D, and E, the different ways in which a pair of objects can be selected while keeping the order into consideration, there is a possibility of 20 different outcomes and each of them is called a permutation. The permutation of five objects, when taken two at a time, can be denoted by 5P2 which is read as “5 Permute 2”. The formula for its evaluation is:

nPk = n!/(n − k)! The expression n!—is read as “n factorial”

Also Read: LCM and HCF for Competitive Exams

Difference between Permutation & Combination

There are some basic differences between Permutation & combination. Look at the table below to know about the difference. 

Difference	Permutation	Combination
Application	Arranging people, digits, numbers, alphabets, numbers, letters & colors	Selection of menu, food, clothes, teams
Keyword	Arrange, Arrangement	Select, Choice
Order	Order matters	Order doesn’t matter
Used for	For Lists	For Groups
Denoted by	nPr	nCr

Calculating Permutations With Ease

While dealing with permutation and combination questions, it is important to practice well in advance so as to solve the problems in less duration. For that, you need to remember the formulas and use them besides formulating other shortcuts to go about solving questions in this section to save time. Here is the easiest way to solve questions on permutations. 

If we have to know how many permutations exist for a given set of numbers-1, 2, 3, 4 without listing them, then the one way of going about it is using the Multiplication Method of Combinatorics in which we multiply the given numbers. 

4×3×2×1=24

Thus, the above set of given numbers has 24 permutations. 

But suppose, we want to know how many permutations does the above set of numbers have when repetition is included such as 5555 and 4444. To evaluate the answer, we can multiply the bigger number, which is 4 as many times as the total number of digits. 

4×4×4×4 = 4⁴ = 256

Example: How many different ways the letters ‘House’ can be arranged?

The word ‘House’ contains 5 letters

Thus, the required number of words using formula ⁵P₅

                           =5!

                           =(1×2×3×4×5) 

                           =120

Also Read: Blood Relations Questions For Competitive Exams

Calculating Combinations With Ease

Combinations are comparatively easier and do not require an order. The formula to evaluate the combination of any given set of numbers is:

nCr = n!/r! * (n – r)! where n represents the total number of items, and r represents the number of items that are chosen at a time.

In how many ways can a coach choose three players from among five players?

There are 5 players to be taken 3 at a time.

Using the formula:

C(5,3) = P(5,3)/ 3! 

           = 5×4×3/3×2×1

           =10 

Thus, the coach can choose the players in 10 different ways.

Practice Questions

If you think you know how to solve permutation & combination questions, try your hand at these practice questions. These questions are taken from the previous year’s competitive & entrance exam question papers. 

1. The number of seven digits integers with the sum of digits equal to 10 & formed by digits 1, 2 & 3 only is
   • 55
   • 66
   • 77
   • 88

2. Let S ={1, 2, 3, 4 } the total number of unordered pairs of disjoint subsets of S equals to
   • 25
   • 34
   • 42
   • 41

3. The total number of ways in which 5 balls of different colors can be distributed among 3 persons so that each person gets at least one ball is
   • 75
   • 150
   • 210
   • 243

4. Six cards & six envelopes are numbered 1, 2, 3, 4, 5, 6, and cards are to be placed in envelopes contains exactly one card & no card is placed in the envelope bearing the same number & moreover the card numbered 1 in always placed in enveloped number 2. Then the numbers of ways it can be done is 
   • 264
   • 265
   • 53
   • 67

5. A debate team consists of 6 girls & 4 boys. A team of 4 members is to be selected from this club including the captain (from among these four members) for the team. If the team has to include at most one boy the number of ways selecting the team is
   • 380
   • 320
   • 260
   • 95

Formulas for Permutation & Combinations

It is important to remember the formulas for permutations & combinations. Here are the formulas:

Permutation or Combination	Repetition	Formula
Permutation	Yes	P (n,r) = nr
Permutation	No	P (n,r) =  n! / (n – r)!
Combination	Yes	C (n,r) = n! / r! ( n-r)!
Combination	No	C (n+ r -1 ,r) = (n + r -1 )! / r! (n – 1) !

Books for Permutation & combinations

There are unlimited books for practicing & learning about permutations & combinations. We have listed the books famous among students & all the books are available easily online & in local markets. These books will help you in revision & practice purposes.

• Permutation and Combinations Chandra Ramesh
• Essential Permutations & Combinations: A Self-teaching Guide
• Probability: Mastering Permutations and Combinations
• Combinatorics of Permutations
• Principles and Techniques in Combinatorics

Permutation and Combination: Things to Remember

• A permutation is to be used if a problem calls for the number of arrangements of objects and different orders is to be counted.
• If a problem calls for the number of ways of selecting objects and the order of selection is not to be counted then a Combination is to be used.
• To ace any competitive exam, it is imperative to maintain a strategy for each section while going forward with your preparation. We recommend you, not to memorize the formulas, but understand how they work.

FAQs

Also Read: How to Prepare for Competitive Exams

Hopefully, this blog has helped you understand the concept of permutation and combination for competitive exams. If you’re looking forward to preparing for the quantitative section for exams like GMAT, GRE, etc. reach out to us at Leverage Edu and we can provide you with the best study preparation guide. All the Best!