6. Permutations and Combinations – Class 11 Mathematics

Introduction

Counting is one of the most basic and yet powerful ideas in mathematics. Whenever we are faced with a situation where we need to find the number of ways something can happen, we turn to the tools of permutations and combinations.

This chapter introduces the methods that allow us to calculate the total number of possible arrangements and selections under different conditions. The journey begins with the Fundamental Principle of Counting (FPC), which gives us the backbone for solving all advanced problems. From there, we move to permutations, which deal with arrangements, and combinations, which focus on selections.

Why is this important?

• For competitive exams like JEE Main, JEE Advanced, KCET, and COMEDK, permutations and combinations form the foundation for probability, binomial theorem, and advanced counting problems.
• For board exams, this chapter contributes several direct problems, often ranging between 6 to 8 marks.
• In real life, these principles apply to situations like arranging students in a line, forming committees, creating passwords, and even predicting outcomes in card games.

In short, mastering this chapter equips you with essential problem-solving skills for both academics and real-world reasoning.

6.2 Fundamental Principle of Counting

The Fundamental Principle of Counting (FPC) helps us calculate outcomes for multi-step processes.

Rule of Multiplication

If a process can be done in m ways and another in n ways, then both processes together can be completed in m × n ways.

Example:

• If you can choose a shirt in 3 ways and trousers in 2 ways, then the total number of outfits = 3 × 2 = 6.

Rule of Addition

If there are two tasks that cannot be performed simultaneously, and the first can be done in m ways while the second can be done in n ways, then either of them can be done in m + n ways.

Example:

• If you can travel to school by bus in 2 ways or by bicycle in 1 way, then total = 2 + 1 = 3 ways.

General Form

If a process has k steps, with options: n₁, n₂, …, nk at each step, then total outcomes = n₁ × n₂ × … × nk.

Illustration for Exams:
Number of 4-digit numbers that can be formed from digits 1 to 9 (repetition allowed) = 9 × 9 × 9 × 9 = 6561.

The FPC is simple but underpins everything in this chapter.

6.3 Permutations

A permutation refers to the arrangement of objects in a specific order. The order of selection matters here.

Formula

The number of permutations of n distinct objects taken r at a time is:

nPr = n! / (n – r)!

Where:

• n = total objects
• r = objects chosen
• ! = factorial function

Types of Permutations

1. Without Repetition
   Each object is used once.
   Example: Number of ways to arrange 3 letters A, B, C = 3! = 6.

2. With Repetition
   Objects can repeat.
   Example: Number of 3-digit numbers using digits 1, 2, 3 = 3³ = 27.

3. Circular Permutations
   Arrangements around a circle are different from linear arrangements.
   Formula: (n – 1)! for n objects.
   Example: Seating 5 people around a round table = 4! = 24 ways.

4. Permutations with Identical Objects
   When some objects are repeated, divide by factorials of their counts.
   Example: Word “BALLOON” = 7! / (2! × 2! × 1! × 1! × 1!) = 1260.

6.4 Combinations

A combination is a selection of objects without considering order.

Formula

The number of combinations of n objects taken r at a time is:

nCr = n! / [r!(n – r)!]

Properties of Combinations

1. Symmetry: nCr = nC(n – r)
2. Boundary: nC0 = nCn = 1
3. Pascal’s Identity: nCr + nC(r – 1) = (n + 1)Cr

Examples

• Choosing 2 players from 5 = 5C2 = 10.
• Selecting 11 players from 15 cricketers = 15C11 = 1365.

Difference between Permutations and Combinations

• Permutations: Order matters (e.g., arranging books).
• Combinations: Order doesn’t matter (e.g., choosing team members).

Real-Life Applications

1. Passwords and PIN codes – Permutations with/without repetition.
2. Team selections – Combinations, since order doesn’t matter.
3. Seating arrangements – Permutations and circular permutations.
4. Card games – Combinations when choosing cards.
5. Probability theory – Every probability calculation depends on counting techniques.

Common Mistakes

• Confusing permutations and combinations.
• Forgetting to divide by factorials for identical objects.
• Ignoring restrictions in problems (e.g., digits must be even/odd, repetition not allowed).
• Applying the permutation formula when order doesn’t matter.

Exam Weightage

• CBSE Boards: 6–8 marks, usually one short and one long problem.
• JEE Main: 2–3 questions (~8–12 marks), often linked to probability.
• JEE Advanced: 1–2 questions, more conceptual.
• KCET/COMEDK: 2–3 direct formula-based MCQs.

Practice Examples

1. How many 4-digit numbers can be formed using digits 1, 2, 3, 4 if repetition is not allowed?
   Solution: 4P4 = 24.

2. How many ways can a committee of 3 be formed from 10 people?
   Solution: 10C3 = 120.

3. How many circular permutations of 6 people can be made?
   Solution: (6 – 1)! = 120.

4. In how many ways can letters of the word “MISSISSIPPI” be arranged?
   Solution: 11! / (4! × 4! × 2! × 1!) = 34,650.

FAQs

Q1. What is the difference between permutation and combination?
Permutation cares about order; combination does not.

Q2. What formula should I use when repetition is allowed?
Use the multiplication rule directly (nr).

Q3. Is circular arrangement a permutation or combination?
It is a special case of permutation.

Q4. How important is this topic for JEE?
Very important. It links directly to probability, a high-weightage topic.

Summary

• FPC gives the foundation of counting.
• Permutations = arrangements, where order matters.
• Combinations = selections, where order doesn’t matter.
• Advanced cases include repetition, circular permutations, and identical objects.
• This chapter connects strongly with probability, binomial theorem, and algebra.