NCERT SOLUTIONS CLASS 11 MATHS CHAPTER 6 Permutations and Combinations

NCERT Solutions For Class 11 Maths Chapter 6 Permutations and Combinations | MathsOrbit

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Learning Objectives - Chapter 6: Permutations and Combinations

Understand fundamental principle of counting
Differentiate between permutations and combinations
Calculate permutations and combinations using formulas
Solve problems with restrictions and conditions
Understand circular permutations
Apply permutations and combinations to real-world problems

Important Concepts - Permutations and Combinations

nPr

Permutations

ⁿPᵣ = n!/(n-r)!

Arrangements matter

nCr

Combinations

ⁿCᵣ = n!/r!(n-r)!

Selections only

Factorial

n! = n×(n-1)×...×1

0! = 1

↻

Circular Permutations

(n-1)! arrangements

No fixed reference

Key Formulas and Properties:

Fundamental Principle: If event A has m ways, event B has n ways, then A and B together have m×n ways

Permutation: ⁿPᵣ = n!/(n-r)! = n×(n-1)×...×(n-r+1)

Combination: ⁿCᵣ = n!/r!(n-r)! = ⁿPᵣ/r!

ⁿCᵣ = ⁿCₙ₋ᵣ (Complementary combination)

ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's identity)

Circular permutation: (n-1)!

Exploring Permutations and Combinations!

Welcome to Chapter 6: Permutations and Combinations! This chapter introduces you to the fascinating world of counting and arrangements. You'll learn systematic methods to count the number of ways events can occur without actually listing all possibilities.

We'll explore the fundamental principle of counting, learn to distinguish between permutations (where order matters) and combinations (where order doesn't matter), and master the formulas for calculating them. You'll also learn about circular permutations, permutations with restrictions, and applications in probability and real-world scenarios.

Permutations and combinations are fundamental in probability theory, statistics, computer science, and many other fields. Mastering these concepts will help you solve complex counting problems efficiently and understand the mathematics behind various real-world situations!

Fundamental Principle

Multiplication rule

nPr

Permutations

Order matters

nCr

Combinations

Selection only

↻

Circular Arrangements

Round table problems

NCERT Textbook Exercises & Solutions

Complete step-by-step solutions for NCERT textbook exercises

Exercise 6.1 - Fundamental Principle of Counting

1. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of digits is allowed?
(ii) repetition of digits is not allowed?

Solution Steps:

A 3-digit number has hundreds, tens, and units places
(i) When repetition is allowed:
Hundreds place: 5 choices (1,2,3,4,5)
Tens place: 5 choices (repetition allowed)
Units place: 5 choices (repetition allowed)
Total numbers = 5 × 5 × 5 = 125
(ii) When repetition is not allowed:
Hundreds place: 5 choices
Tens place: 4 choices (one digit already used)
Units place: 3 choices (two digits already used)
Total numbers = 5 × 4 × 3 = 60

(i) 125 numbers
(ii) 60 numbers

2. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Solution Steps:

For a number to be even, the units digit must be even
Available even digits: 2, 4, 6 (3 choices)
Units place: 3 choices (must be even)
Hundreds place: 6 choices (any digit 1-6)
Tens place: 6 choices (repetition allowed)
Total even numbers = 6 × 6 × 3 = 108

108 even numbers

Exercise 6.2 - Permutations

1. Evaluate: (i) 8! (ii) 4! - 3!

Solution Steps:

(i) 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40320
(ii) 4! - 3! = (4 × 3 × 2 × 1) - (3 × 2 × 1)
= 24 - 6 = 18

(i) 40320
(ii) 18

2. Is 3! + 4! = 7!?

Solution Steps:

Calculate 3! = 3 × 2 × 1 = 6
Calculate 4! = 4 × 3 × 2 × 1 = 24
Calculate 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
3! + 4! = 6 + 24 = 30
30 ≠ 5040
Therefore, 3! + 4! ≠ 7!

No, 3! + 4! = 30 and 7! = 5040, so they are not equal

📜 Historical Fact: Did you know that the concept of permutations and combinations dates back to ancient India? Indian mathematician Bhaskaracharya (c. 1150 AD) wrote about permutations and combinations in his work "Lilavati." The factorial notation (n!) was introduced by French mathematician Christian Kramp in 1808. The formulas we use today were largely developed in the 17th century!

🎯 Practical Activity: Create a "Password Combinations" experiment! Choose a 4-digit PIN and calculate how many possible combinations exist (with and without repetition). Then try different scenarios: PIN with only even digits, PIN with no repeated digits, etc. This shows how permutations and combinations apply to cybersecurity and password strength!

20 Additional Practice Questions

Extra questions to master Permutations and Combinations Concepts

Multiple Choice Questions

The value of ⁷P₂ is:
A) 14 B) 42 C) 21 D) 7
If ⁿC₈ = ⁿC₂, then n is:
A) 2 B) 8 C) 10 D) 6
The number of ways to arrange 5 different books on a shelf is:
A) 25 B) 120 C) 5 D) 10
⁶C₂ + ⁶C₃ is equal to:
A) ⁷C₃ B) ⁷C₄ C) ⁶C₅ D) ⁸C₃
The number of diagonals in a hexagon is:
A) 6 B) 9 C) 12 D) 15

Fill in the Blanks

0! = _______.
The number of permutations of n distinct objects taken all at a time is _______.
If ⁿPᵣ = 720 and ⁿCᵣ = 120, then r = _______.
The number of ways to select 2 students from 10 is _______.
ⁿCᵣ = _______ when r = n.

True or False

ⁿPᵣ is always greater than ⁿCᵣ for r > 1. (True/False)
5! = 120. (True/False)
ⁿCₙ = 1. (True/False)
The number of ways to arrange 5 people in a circle is 5!. (True/False)
⁶C₂ = ⁶C₄. (True/False)

Short Answer Questions

Find n if ⁿP₄ = 20 × ⁿP₂
How many words can be formed using all letters of the word "MATHS"?
In how many ways can 5 boys and 3 girls be seated in a row so that no two girls are together?
Find the number of triangles that can be formed from 8 points on a circle.
If ⁿC₉ = ⁿC₈, find ⁿC₁₇.

Answer Key for Practice Questions

Multiple Choice:

1. B) 42
2. C) 10
3. B) 120
4. A) ⁷C₃
5. B) 9

Fill in the Blanks:

6. 1
7. n!
8. 3
9. 45
10. 1

True or False:

11. True
12. True
13. True
14. False (it's 4!)
15. True

Short Answers:

16. n = 7
17. 120 words
18. 14400 ways
19. 56 triangles
20. 1

Chapter Summary - Permutations and Combinations

Key Concepts:

Fundamental Principle of Counting:

If one operation can be performed in m ways and another in n ways, then both operations can be performed in m × n ways
Also known as multiplication principle
Applies to sequential independent events

Factorial Notation:

n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1
0! = 1 (by definition)
1! = 1
n! = n × (n-1)!

Permutations:

Arrangements where order matters
ⁿPᵣ = n!/(n-r)! = n × (n-1) × ... × (n-r+1)
ⁿPₙ = n! (permutations of n objects taken all at a time)
ⁿP₀ = 1

Types of Permutations:

Permutations with Repetition:

When some objects are identical
Number of permutations = n!/(p! × q! × r! × ...)
Where p, q, r,... are numbers of identical objects

Circular Permutations:

Arrangements around a circle
Number of arrangements = (n-1)!
No fixed reference point
For clockwise and anticlockwise same: (n-1)!/2

Combinations:

Selections where order doesn't matter
ⁿCᵣ = n!/[r!(n-r)!]
ⁿCᵣ = ⁿCₙ₋ᵣ (complementary combination)
ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's identity)
ⁿC₀ = ⁿCₙ = 1

Important Properties:

Relationship between P and C:

ⁿPᵣ = ⁿCᵣ × r!
ⁿCᵣ = ⁿPᵣ / r!
Permutations count arrangements, combinations count selections

Special Cases:

Number of diagonals in n-sided polygon = ⁿC₂ - n
Number of handshakes among n people = ⁿC₂
Number of straight lines from n points (no three collinear) = ⁿC₂
Number of triangles from n points (no three collinear) = ⁿC₃

Teaching Guide & Notes

📚 Effective Teaching Strategies:

Use real-life examples to explain fundamental counting principle
Practice distinguishing between permutations and combinations
Use visual aids for circular permutations
Create hands-on activities with physical objects
Organize group problem-solving sessions

🎯 Common Learning Gaps:

Confusion between permutations and combinations
Difficulty in identifying when to use which formula
Problems with factorial calculations for large numbers
Challenges in solving problems with restrictions

🔍 Key Concepts to Emphasize:

Clear understanding of fundamental counting principle
Mastery of factorial notation and calculations
Proficiency in distinguishing permutations vs combinations
Solid understanding of circular permutations
Ability to solve problems with various restrictions