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Permutations and Combinations

Practice MCQs

Permutations and Combinations are important counting methods used to find the number of possible arrangements or selections.

Quantitative Aptitude Permutations and Combinations Competitive Exams

Permutations and Combinations are important counting methods used to find the number of possible arrangements or selections. These concepts are commonly asked in competitive exams, aptitude tests, probability, seating arrangements, ranking problems, and selection-based questions.

What are Permutations and Combinations?

Permutation means arrangement. It is used when the order of objects is important. For example, arranging students in a row, forming numbers from digits, or arranging letters of a word.

Combination means selection. It is used when the order of objects is not important. For example, selecting students for a team, choosing questions from a paper, or forming a committee.

Quick idea: If order matters, use permutation. If order does not matter, use combination.
Concept Meaning Example
Permutation Arrangement where order matters ABC and BAC are different
Combination Selection where order does not matter ABC and BAC represent the same group
Factorial Product of natural numbers down to 1 \(5! = 5\times4\times3\times2\times1\)
Fundamental Counting Principle Multiply choices at each step 3 shirts and 2 pants give \(3\times2=6\) outfits

“Permutation is arrangement; combination is selection.”

Aptitude Tip
Key points
  • Permutation is used when order matters.
  • Combination is used when order does not matter.
  • Factorial notation is very important.
  • \(0! = 1\) and \(1! = 1\).
  • Use multiplication rule for step-by-step choices.
  • In selection problems, avoid counting duplicate arrangements.
arrangement selection factorial counting

Visual Understanding

These diagrams show the difference between arrangement and selection.

Permutation: Order Matters
A B C Arrangement 1: A B C B A C Arrangement 2: B A C
\[ ABC \neq BAC \]

In permutation, changing the order creates a different arrangement.

Combination: Order Does Not Matter
A B C Selected Group = {A, B, C} ABC, BAC, CAB are the same selection.
\[ \{A,B,C\}=\{B,A,C\} \]

In combination, only the selected group matters, not the order.

Important Formulas and Rules

Factorial
\[ n! = n(n-1)(n-2)\cdots 1 \]

Factorial means product of all positive integers up to \(n\).

Permutation
\[ {}^nP_r = \frac{n!}{(n-r)!} \]

Used for arranging \(r\) objects from \(n\) objects.

Combination
\[ {}^nC_r = \frac{n!}{r!(n-r)!} \]

Used for selecting \(r\) objects from \(n\) objects.

Relation
\[ {}^nP_r = {}^nC_r \times r! \]

Arrange the selected \(r\) objects in \(r!\) ways.

All Objects Arranged
\[ n! \]

Number of arrangements of \(n\) different objects.

Circular Permutation
\[ (n-1)! \]

Used when \(n\) people sit around a circular table.

Repeated Objects
\[ \frac{n!}{p!q!r!} \]

Used when some objects are repeated.

Complement Rule
\[ {}^nC_r = {}^nC_{n-r} \]

Choosing \(r\) is same as rejecting \(n-r\).

Rule: Use permutation for arrangements and combination for selections.

Common Types of Questions

Arrangement Questions

Objects or people are arranged in order.

  • Arranging letters
  • Arranging students
  • Ranking positions
  • Order matters
Selection Questions

Objects or people are selected without order.

  • Committee selection
  • Team formation
  • Choosing questions
  • Order does not matter
Digit Formation

Numbers are formed using given digits.

  • Repeated digits allowed
  • No repetition
  • Even or odd numbers
  • First digit cannot be zero
Word Arrangement

Letters of a word are arranged.

  • Different letters
  • Repeated letters
  • Vowels together
  • Consonants together
Exam approach: First decide whether the question is asking for arrangement or selection. This single decision usually gives the correct method.
Method Bank
Arrange 5 People

All 5 different people in a row:

\[ 5! = 120 \]
Select 3 from 5

Selection, so use combination:

\[ {}^5C_3 = 10 \]
Arrange 3 from 5

Arrangement, so use permutation:

\[ {}^5P_3 = 60 \]
Repeated Letters

For word LEVEL:

\[ \frac{5!}{2!2!} \]

Tip: Selection first, arrangement later. If selected objects must also be ordered, use permutation.

Factorial Concept
5 × 4 × 3 × 2 × 5! = 5 × 4 × 3 × 2 × 1 = 120
Factorial counts the number of ways to arrange all different objects.

Step-by-Step Solving Method

Step Action Example
Step 1 Read the question carefully. Are we arranging or selecting?
Step 2 Check whether order matters. Rank 1, 2, 3 means order matters.
Step 3 Choose formula. Use \(P\) for arrangement, \(C\) for selection.
Step 4 Substitute values. \({}^5P_3=\frac{5!}{2!}\)
Step 5 Simplify carefully. \(5\times4\times3=60\)
Important: In combination questions, different orders of the same selected group are counted only once.

Solved Examples

Question Method Answer
How many ways can 5 different books be arranged on a shelf? All 5 books are arranged.
\[ 5! = 5\times4\times3\times2\times1 = 120 \]
120
How many ways can 3 students be selected from 5 students? Order does not matter, so use combination.
\[ {}^5C_3 = \frac{5!}{3!2!} \]
\[ {}^5C_3 = 10 \]
10
How many ways can first, second, and third rank be given among 5 students? Order matters, so use permutation.
\[ {}^5P_3 = \frac{5!}{(5-3)!} \]
\[ {}^5P_3 = \frac{5!}{2!}=5\times4\times3=60 \]
60
How many ways can the letters of the word CAT be arranged? All letters are different.
\[ 3! = 3\times2\times1 = 6 \]
6
How many ways can the letters of the word LEVEL be arranged? LEVEL has 5 letters. L repeats 2 times and E repeats 2 times.
\[ \frac{5!}{2!2!} \]
\[ \frac{120}{4}=30 \]
30
How many committees of 4 members can be formed from 8 people? Committee is selection, so order does not matter.
\[ {}^8C_4 = \frac{8!}{4!4!} \]
\[ {}^8C_4 = 70 \]
70
How many 3-digit numbers can be formed using digits 1, 2, 3, 4 without repetition? Order matters because different digit orders create different numbers.
\[ {}^4P_3 = 4\times3\times2 \]
24
How many ways can 6 people sit around a circular table? Circular permutation of \(n\) different objects:
\[ (n-1)! \]
\[ (6-1)! = 5! = 120 \]
120

Note: In number formation and ranking problems, order usually matters. In committee and team selection, order usually does not matter.

Common Traps and Shortcuts

Common Traps
  • Using permutation when the question is only about selection.
  • Using combination when the order is important.
  • Forgetting repeated letters in word arrangement problems.
  • Counting the same committee multiple times.
  • Forgetting that first digit cannot be zero in number formation.
  • Using \(n!\) instead of \((n-1)!\) for circular seating.
Useful Shortcuts
  • Arrangement means permutation.
  • Selection means combination.
  • \({}^nC_r = {}^nC_{n-r}\).
  • \({}^nP_r = {}^nC_r \times r!\).
  • For repeated objects, divide by factorials of repetitions.
  • For circular arrangement, fix one object and arrange the rest.
Exam approach: Ask one question first: “Does order matter?” If yes, use permutation. If no, use combination.

Practice

A) Multiple Choice Questions
  1. How many ways can 4 different books be arranged on a shelf?
    12 16 24 36
  2. How many ways can 2 students be selected from 6 students?
    12 15 20 30
  3. Find \({}^5P_2\).
    10 15 20 25
  4. Find \({}^5C_2\).
    5 10 15 20
  5. How many ways can 5 people sit around a circular table?
    24 60 100 120
B) Solve the Higher-Order Problems
  1. How many 3-letter words can be formed from the letters A, B, C, D without repetition? (Hint: Order matters.)
  2. How many committees of 3 members can be formed from 7 people? (Hint: Order does not matter.)
  3. How many ways can the letters of the word APPLE be arranged? (Hint: P repeats twice.)
  4. How many ways can 7 people sit around a circular table? (Hint: Circular arrangement is \((n-1)!\).)
  5. How many 4-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition? (Hint: Arrange 4 from 5.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Permutation Arrangement where order matters
Combination Selection where order does not matter
Factorial Product of natural numbers down to 1
Circular Arrangement Arrangement around a circle
Repeated Objects Objects appearing more than once
Fundamental Counting Principle Multiply number of choices at each step
Aptitude Reminder

Permutations and combinations are counting tools. Use permutation when order is important, and combination when only selection is important. Repeated objects, circular arrangements, and digit restrictions should be handled carefully.

Task: Create five questions using arrangements, selections, word arrangements, circular seating, and number formation.

Show Suggested Answers
Multiple Choice
  1. 24
    \[ 4! = 4\times3\times2\times1 = 24 \]
  2. 15
    \[ {}^6C_2 = \frac{6!}{2!4!}=15 \]
  3. 20
    \[ {}^5P_2 = 5\times4=20 \]
  4. 10
    \[ {}^5C_2 = \frac{5!}{2!3!}=10 \]
  5. 24
    \[ (5-1)! = 4! = 24 \]
Higher-Order Problems
  1. 3-letter words from A, B, C, D without repetition:
    \[ {}^4P_3 = 4\times3\times2 = 24 \]
    Answer = 24.
  2. Committees of 3 from 7 people:
    \[ {}^7C_3 = \frac{7!}{3!4!}=35 \]
    Answer = 35.
  3. APPLE has 5 letters, and P repeats twice:
    \[ \frac{5!}{2!}=\frac{120}{2}=60 \]
    Answer = 60.
  4. 7 people around a circular table:
    \[ (7-1)! = 6! = 720 \]
    Answer = 720.
  5. 4-digit numbers from 5 digits without repetition:
    \[ {}^5P_4 = 5\times4\times3\times2=120 \]
    Answer = 120.
Concept Matching
  1. Permutation → Arrangement where order matters
  2. Combination → Selection where order does not matter
  3. Factorial → Product of natural numbers down to 1
  4. Circular Arrangement → Arrangement around a circle
  5. Repeated Objects → Objects appearing more than once
  6. Fundamental Counting Principle → Multiply number of choices at each step
Clue Explanation

The main difference is order. Arrangement uses permutation; selection uses combination. Repetition and circular arrangements require special formulas.

Exam tips
  • Ask whether order matters.
  • Use permutation for arrangement.
  • Use combination for selection.
  • Use factorial for arranging all objects.
  • For repeated letters, divide by repeated factorials.
  • For circular seating, use \((n-1)!\).
Practice MCQs