Mastering the Permutations Concepts with Examples, Word Problems & Practice Sets

Permutation in Mathematics – Complete Guide

Permutation is a fundamental concept in combinatorics that deals with the arrangement of objects. It answers the question: “In how many ways can we arrange a set of objects?” Whether you’re preparing for competitive exams or just learning for fun, mastering permutations will boost your confidence in solving advanced problems.

“Permutation is not about just counting, it’s about understanding possibilities.”

What is a Permutation?

A permutation refers to an arrangement of all or part of a set of objects. The order in which the elements are arranged is important in permutations.

Permutation Formula

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

Where:

• \( n \) = total number of objects
• \( r \) = number of objects taken at a time
• \( n! \) = factorial of \( n \)

Types of Permutations

🔢 Permutations with Distinct Objects:

📚 Permutations with Repeated Objects:

Used when some objects are repeated.

🎯 Circular Permutations:

Understanding Fundamental Principles in Permutations

📘 Principle of Addition

Definition:
If an event A can occur in \(m\) ways, and another event B can occur in \(n\) ways, and both events are mutually exclusive (i.e., they can't happen at the same time), then the total number of ways either A or B can happen is:

\[ \text{Total ways} = m + n \]

Example:
A student can choose a math book in 3 ways or a science book in 4 ways. Then, the total number of ways to choose one book is:

\[ 3 + 4 = 7 \text{ ways} \]

📗 Principle of Multiplication

Definition:
If an event A can occur in \(m\) ways, and another event B can occur in \(n\) ways, and both events must occur together, then the total number of ways they can occur together is:

\[ \text{Total ways} = m \times n \]

Example:
A person has 3 shirts and 2 trousers. The number of different outfit combinations is:

\[ 3 \times 2 = 6 \text{ combinations} \]

Permutations: With and Without Repetition

🔹 Permutations Without Repetition

Definition:
When all objects are distinct and each object can be used only once in an arrangement, permutations are said to be without repetition.

Key Concept:
Arrange \(r\) objects chosen from \(n\) distinct objects, without repeating any object.

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

Explanation:
- Start with \(n\) choices for the first position.
- Then \(n-1\) choices for the second position (since no repetition).
- Continue decreasing choices until \(r\) positions are filled.
- Multiply these choices: \(n \times (n-1) \times \cdots \times (n-r+1)\), which equals \(\frac{n!}{(n-r)!}\).

Example:
How many 3-digit numbers can be formed using digits 1 to 5 without repetition?

Solution:
\[ {}^5P_3 = \frac{5!}{(5-3)!} = \frac{120}{2} = 60 \]

So, 60 different numbers can be formed.

🔸 Permutations With Repetition

Definition:
When some objects repeat and are indistinguishable, or when repetition of objects is allowed in the arrangement.

Two main cases:

1. Repeated objects in arrangement: Some objects are identical, so total permutations divide by factorials of repetitions.
   Formula: \[ \frac{n!}{p! \times q! \times \cdots} \] where \(p, q, \dots\) are counts of repeated objects.
2. Repetition allowed (same objects can be reused): For arranging \(r\) objects each chosen from \(n\) distinct objects with repetition,
   Formula: \[ n^r \] since each position has \(n\) choices.

Explanation of first case:
Since some objects are identical, swapping them does not create a new unique arrangement. So we divide total arrangements by the factorial of the counts of identical objects.

Example (repeated letters):
Number of distinct ways to arrange letters of “BALLOON” where L and O repeat:

Solution:
Total letters \(n = 7\), repeated: L(2), O(2), others distinct.
\[ \text{Permutations} = \frac{7!}{2! \times 2!} = \frac{5040}{4} = 1260 \]

Explanation of second case (repetition allowed):
When repetition is allowed, each of the \(r\) positions can be chosen independently from \(n\) objects.
So total permutations are \(n \times n \times \cdots\) (r times) = \(n^r\).

Example (repetition allowed):
How many 3-letter words can be formed using letters A, B, C if repetition is allowed?

Solution:
Each position: 3 choices.
\[ 3^3 = 27 \]

So, 27 such words can be formed.

Permutation Examples with Detailed Explanations

Example 1: Simple Arrangement of Students

Problem:
How many ways can 4 students be arranged in a row?

Explanation:
Arrange 4 distinct students in sequence. Number of permutations of \(n\) distinct objects is:

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

Calculation:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Answer: 24 ways.

Example 2: Permutations of Subsets

Problem:
How many 3-letter words can be formed using the letters of the word “MATH”?

Explanation:
Number of ways to arrange \(r=3\) letters chosen from \(n=4\) distinct letters is:

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

Calculation:
\[ {}^4P_3 = \frac{4!}{(4-3)!} = \frac{24}{1} = 24 \]

Answer: 24 ways.

Example: Types of Yearly Calendars

Problem:
How many different types of calendars are possible for a year?

Explanation:
A calendar type depends on:

• The day of the week the year starts on (7 possible starting days)
• Whether it’s a leap year or a non-leap year (2 types)

So, we multiply the two possibilities:

\[ \text{Total Types} = 7 \times 2 = 14 \]

Answer: 14 types of yearly calendars.

Example: Permutation of Athletes

Problem:
There are 12 athletes running a race. In how many different ways can the top 3 positions (1st, 2nd, and 3rd) be awarded?

Explanation:
Since the order of winning matters, we use the permutation formula:
Number of ways to arrange \( r = 3 \) positions from \( n = 12 \) athletes is:

\[ {}^{12}P_3 = \frac{12!}{(12 - 3)!} \]

Calculation:
\[ {}^{12}P_3 = \frac{12!}{9!} = 12 \times 11 \times 10 = 1320 \]

Answer: 1320 possible ways for the athletes to win the top 3 positions.

Example: Flag Signal Arrangements

Problem:
There are 5 different flags. If 2 flags are used at a time to give a signal, how many different signals are possible?

Explanation:
Since each flag is of a different color and order matters, this is a case of permutation of 2 from 5:

\[ {}^5P_2 = \frac{5!}{(5 - 2)!} \]

Calculation:
\[ {}^5P_2 = \frac{5!}{3!} = \frac{120}{6} = 20 \]

Answer: 20 different flag signals are possible.

Why Allowing Repetition Increases Count?

When digits can repeat, each position becomes independent, allowing more flexible arrangements, greatly increasing the total.

Example: Count of Odd Three-Digit Numbers

Problem:
How many odd three-digit numbers are there between 100 and 999?

Explanation:
A three-digit number has 3 places: Hundreds, Tens, and Units.
We want the number to be odd, so the units digit must be 1, 3, 5, 7, or 9 (5 options).

• Hundreds place (A): 1 to 9 → 9 choices
• Tens place (B): 0 to 9 → 10 choices
• Units place (C): only odd digits → 5 choices (1, 3, 5, 7, 9)

Calculation:

\[ 9 \times 10 \times 5 = 450 \]

Answer: 450 odd three-digit numbers exist between 100 and 999.

Shortcut Insight:

Out of every 2 consecutive numbers, one is odd. So half of the 900 three-digit numbers are odd: \( \frac{900}{2} = 450 \) – same result!

Example: Using {3, 5, 6, 7, 8} — Repetition Allowed

Problem:
How many four-digit numbers can be formed using digits {3, 5, 6, 7, 8} such that:

• The number is greater than 5000
• The number is odd
• Repetition of digits is allowed

Explanation:
Let's form a number ABCD:

• A (thousands place): Must be ≥ 5 → options: 5, 6, 7, 8 → 4 choices
• B and C: Any of the 5 digits (repetition allowed) → 5 × 5 = 25 choices
• D (units place): Must be odd → 3, 5, 7 → 3 choices

Calculation:

\[ 4 \times 5 \times 5 \times 3 = 300 \]

Answer: 300 valid four-digit numbers.

Example 3: Permutations with Repeated Letters

Problem:
How many distinct ways to arrange letters of “LEVEL”?

Explanation:
Total letters \(n=5\), repeated letters: L (2 times), E (2 times). Use formula:

\[ \text{Permutations} = \frac{n!}{p! \times q! \times \cdots} \]

Calculation:
\[ \frac{5!}{2! \times 2!} = \frac{120}{2 \times 2} = 30 \]

Answer: 30 ways.

Example 4: Circular Permutations

Problem:
How many ways can 5 people be seated around a round table?

Explanation:
For circular permutations, one position is fixed to remove identical rotations. Number of ways:

\[ (n-1)! = (5-1)! = 4! = 24 \]

Answer: 24 ways.

Example 5: Permutations of Letters with Multiple Repetitions

Problem:
How many distinct ways to arrange letters of “MISSISSIPPI”?

Explanation:
Total letters \(n=11\), repeated letters: M(1), I(4), S(4), P(2). Use formula:

\[ \frac{11!}{1! \times 4! \times 4! \times 2!} \]

Calculation:
Calculate factorial values:
\[ 11! = 39916800,\quad 4! = 24,\quad 2! = 2 \] \[ \Rightarrow \frac{39916800}{1 \times 24 \times 24 \times 2} = \frac{39916800}{1152} \] \[= 34650\]

Answer: 34,650 distinct arrangements.

Example 6: Selecting and Arranging Books

Problem:
From 8 distinct books, how many ways to select and arrange 3 on a shelf?

Explanation:
Select \(r=3\) books from \(n=8\), then arrange. Use:

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

Calculation:
\[ {}^8P_3 = \frac{8!}{5!} = \frac{40320}{120} = 336 \]

Answer: 336 ways.

Example 7: Word Problem - Lock Combinations

Problem:
A lock uses a 4-digit code from digits 0-9, no repetition allowed. How many possible codes?

Explanation:
Choose and arrange \(r=4\) digits out of \(n=10\) distinct digits.

\[ {}^{10}P_4 = \frac{10!}{6!} = 10 \times 9 \times 8 \times 7 = 5040 \]

Answer: 5040 possible codes.

Example 8: Arranging People in a Photo

Problem:
From 6 friends, how many ways to arrange 4 in a line for a photo?

Explanation:
Choose \(r=4\) from \(n=6\) and arrange:

\[ {}^6P_4 = \frac{6!}{2!} = \frac{720}{2} = 360 \]

Answer: 360 ways.

Example 9: Arranging Flags

Problem:
How many ways to arrange 7 different flags on a pole?

Explanation:
Arrange all 7 distinct flags:

\[ 7! = 5040 \]

Answer: 5040 ways.

Example 10: Forming Number Plates

Problem:
Number plates have 3 letters followed by 3 digits. Letters and digits cannot repeat. How many unique plates?

Explanation:
Choose and arrange 3 letters from 26, then 3 digits from 10:

\[ {}^{26}P_3 \times {}^{10}P_3 = \frac{26!}{23!} \times \frac{10!}{7!} = (26 \times 25 \times 24) \times (10 \times 9 \times 8) = 15600 \times 720 = 11,232,000 \]

Answer: 11,232,000 unique plates.

Example 11: Arranging Students in Groups

Problem:
From 10 students, 5 stand in a line and 5 stand behind them. How many arrangements possible?

Explanation:
Arrange 5 students in front and remaining 5 behind. Both lines permutations independent.

\[ 5! \times 5! = 120 \times 120 = 14400 \]

Answer: 14,400 arrangements.

🧠 Word Problems Involving Addition and Multiplication Principles

Example 1: Competition Selection

Problem: A student can qualify for a math competition in 3 different ways or a science competition in 2 different ways. Once selected, they must choose 1 topic from 4 available topics to prepare for.

Explanation:
- First, we have **two mutually exclusive options**: math or science competition.
- Apply the Addition Principle:
\[ \text{Ways to qualify} = 3 + 2 = 5 \] - Then, for each qualification, they must select 1 of 4 topics:
- Apply the Multiplication Principle:
\[ \text{Total options} = 5 \times 4 = 20 \text{ ways} \]

Example 2: Subject and Code Creation

Problem: A student can choose either 2 foreign languages or 3 electives. After choosing a subject, they must create a 3-letter code from its letters.

Explanation:
- First, choose 1 subject from 5 total options (2 + 3). These are **alternative choices**, so:
\[ 2 + 3 = 5 \text{ subject choices} \] - Then, for each, a 3-letter code is formed (without repetition):
\[ {}^5P_3 = \frac{5!}{(5-3)!} = 60 \text{ codes per subject} \] - Total codes using the Multiplication Principle:
\[ 5 \times 60 = 300 \text{ possible codes} \]

Example 3: Café Orders

Problem: A café offers 3 coffee types and 2 tea types. After selecting a drink, a customer can add 1 of 5 available flavors. How many unique order combinations are possible?

Explanation:
- Choosing a drink: 3 coffee + 2 tea = 5 options (Addition Principle)
\[ 3 + 2 = 5 \text{ drink options} \] - For each drink, 5 flavor choices (Multiplication Principle):
\[ 5 \times 5 = 25 \text{ unique orders} \]

Example 4: Travel Routes

Problem: A person can travel to a city via 2 bus routes, 3 train routes, or 1 flight. Once in the city, they can take any of 4 taxis to reach their hotel.

Explanation:
- Travel to the city (Addition):
\[ 2 + 3 + 1 = 6 \text{ travel-to-city options} \] - Travel within the city (Multiplication):
\[ 6 \times 4 = 24 \text{ total travel plans} \]

Example 5: Password Creation

Problem: A user can create a password in one of two ways: (a) Choose 3 letters from A–Z, or (b) Choose 2 digits (0–9) and 1 special symbol from (!, @, #). How many different passwords are possible?

Explanation:
Option A: Choose 3 letters from 26 without repetition:
\[ {}^{26}P_3 = 26 \times 25 \times 24 = 15600 \]
Option B: Choose 2 digits from 10 and 1 symbol from 3:
\[ 10 \times 9 \times 3 = 270 \]
- Since either option A or B is allowed (Addition Principle):
\[ 15600 + 270 = 15870 \text{ passwords} \]

❌ Common Mistakes to Avoid

• Not considering repeated elements in the arrangement
• Confusing permutation with combination (order matters in permutation!)
• Forgetting to subtract when selecting objects (\(n - r\))

📍 Where You’ll Use Permutations

• Probability problems
• Cryptography & password generation
• Seating and event arrangements
• Genetics & DNA sequence modeling
• Game theory and logic puzzle solving

🎯 Permutation Practice Questions by Type

🔢 Basic Permutation Problems

1. In how many ways can 6 books be arranged on a shelf?
2. Arrange 5 people in a line.
3. How many ways to arrange the digits 2, 3, 4, 5?
4. Find the number of ways to arrange 8 different flags in a row.
5. Arrange 7 paintings on a wall.
6. Arrange 9 different students in a row for a photo.
7. How many ways to assign 4 different prizes to 4 winners?
8. Arrange the digits 1 through 5 without repetition.

🔁 Permutations with Repetition

1. How many 3-letter words can be made from A, B, C if repetition is allowed?
2. Form 4-digit numbers using 0–9 (repetition allowed).
3. How many 5-letter codes can be formed from 26 letters?
4. Arrange digits 1 to 3 in 3 places with repetition.
5. Create 2-digit numbers from 1–5 with repetition.
6. Passwords with 2 letters followed by 2 numbers (with repetition).
7. Form license plates using 3 letters + 2 digits (allow repetition).
8. Arrange 4 colors from a set of 6 with repetition.

📚 Permutations with Repeated Objects

1. How many ways to arrange “BALLOON”?
2. Find arrangements of “SUCCESS”.
3. Arrange the letters of “LEVEL”.
4. How many ways to write “COMMITTEE”?
5. Arrange “MISSISSIPPI”.
6. How many arrangements of “TATTOO”?
7. How many ways to write “STATISTICS”?
8. Arrange the word “ENGINEERING”.

⭕ Circular Permutations

1. Arrange 6 people around a round table.
2. In how many ways can 5 boys sit in a circle?
3. Arrange 4 friends in a circle.
4. Arrange 7 items in a circle (clockwise).
5. How many necklaces can be made using 5 beads?
6. In how many ways can 3 couples sit at a round table with alternate seating?
7. Arrange 8 children in a circular dance.
8. How many circular permutations of 10 flags?

🚫 Permutations with Restrictions

1. Arrange 5 people in a row if A and B must sit together.
2. Arrange 6 students where X cannot sit at the end.
3. Arrange “DELHI” such that vowels are together.
4. Arrange 7 books so that math books are not together.
5. Arrange 5 girls and 3 boys so boys don’t sit together.
6. In how many ways can “APPLE” be arranged so that P's are together?
7. Arrange 6 seats such that 2 fixed people are not adjacent.
8. How many ways to seat 4 people so that A is always before B?

✍️ Word-Based Permutations

1. Arrange the letters of “FRIEND”.
2. How many arrangements of “BANANA”?
3. Arrange “PARALLEL”.
4. Arrange “HISTORY” starting with H.
5. Arrange “ENGLISH” with vowels in the middle.
6. Arrange “TRIANGLE” ending in E.
7. Arrange “PROBLEM” where B is not first.
8. Arrange “CALCULATOR” with all vowels together.

🌍 Real-Life Permutations

1. Create passwords of 3 letters and 2 digits (no repetition).
2. Assign 3 roles (President, VP, Secretary) to 6 students.
3. Choose team order of 4 runners from 8 athletes.
4. How many PINs of 4 digits (non-repeating) are possible?
5. Arrange product IDs using 3 digits + 2 letters.
6. Seat 4 couples around a table alternating genders.
7. Assign 5 different tasks to 5 people.
8. Form different locker codes using 5 symbols.

🔄 Selection & Arrangement

1. Select and arrange 3 players out of 6.
2. Select 4 students and seat them in a row.
3. Choose 2 letters from A–Z and arrange them.
4. Choose 3 digits from 0–9 and arrange them.
5. Select 3 books from 5 and arrange on a shelf.
6. Choose and assign medals to top 3 of 10 athletes.
7. Select 5 flags out of 8 and arrange them on a pole.
8. Select 4 workers from 7 and assign them to 4 jobs.

“Permutation isn't just about numbers, it's about patterns, logic, and strategy.”

Final Thoughts

Understanding permutations opens up a powerful world of problem-solving in mathematics. Whether you're preparing for entrance exams, Olympiads, or just want to build logical thinking, permutation problems help sharpen your brain.

Explore more math topics, tricks, and quizzes at Learn4Math.