11th Class Maths Chapter 7 Notes – Solution | Get Now
11th Class Maths Chapter 7 Notes – Solution cover one of the most scoring chapters in the FSc Part 1 mathematics syllabus: Permutation, Combination and Probability. This chapter is heavily based on formulas, so once a student understands the pattern, most questions become quick to solve.
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Many students find this chapter tricky at first because it introduces new symbols like nPr, nCr, and factorial notation. The solved notes break down every exercise step by step so nothing feels confusing.
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You can also check [internal link] for 11th Class Maths Chapter 6 Notes to revise Sequences and Series before starting this chapter.
Factorial Notation Explained
A factorial, written as n!, means multiplying a positive integer by every whole number below it down to 1.
n! = n(n − 1)(n − 2) ……. 3.2.1
Exercise 7.1 focuses entirely on evaluating factorial expressions and writing numbers in factorial form. The 11th Class Maths Chapter 7 Notes – Solution explain each simplification clearly, including tricky fractions like 11!/(4!7!).
Key Factorial Rule
- n! = n(n − 1)! and 0! = 1
This single rule is used repeatedly throughout the chapter, especially in permutation and combination formulas.
Fundamental Principle of Counting
This principle states that if event A can happen in p ways and event B can happen in q ways after A occurs, then both events together can happen in p × q ways.
This concept is the foundation for understanding permutations, and it is explained with simple examples before moving into formulas.
Permutation Explained
A permutation is an arrangement of objects in a specific order. Unlike combination, order matters here.
nPr Formula
The formula for permutation of n objects taken r at a time is:
nPr = n! / (n − r)!
Exercise 7.2 uses this formula to solve problems involving digits, letters, and word arrangements. The 11th Class Maths Chapter 7 Notes – Solution also cover proofs like nPr = n × (n−1)Pr−1, which frequently appear in board exams.
Common Permutation Word Problems
Exercise 7.2 and 7.3 include real exam-style questions such as:
- Forming numbers from given digits without repetition
- Arranging books on a shelf under certain conditions
- Seating boys and girls alternately
- Finding numbers greater than a given value
These word problems are the reason students specifically search for solved notes instead of just formulas.
Permutation of Things Not All Different
When some objects repeat, a different formula is used:
n! / (n1! × n2! × n3!)
This is applied to words like PAKPATTAN, MATHEMATICS, and ASSASSINATION, where letters repeat multiple times.
Circular Permutation
Circular permutation deals with arranging objects around a circle, such as people at a round table or keys on a ring. Unlike straight-line arrangement, one position is fixed to avoid counting rotations as different arrangements.
Exercise 7.3 also covers necklace and key-ring problems, which use a slightly modified circular formula.
Combination Explained
A combination is a selection where order does not matter. This is different from permutation, and mixing the two up is the most common mistake students make.
nCr Formula
nCr = n! / [r!(n − r)!]
Also, an important identity used often is:
nCr = nCn−r
Exercise 7.4 covers finding values of n and r, along with committee formation and geometry-based questions like counting diagonals and triangles in a polygon.
Real-Life Combination Questions
The 11th Class Maths Chapter 7 Notes – Solution include practical examples such as:
- Forming committees with specific numbers of men and women
- Selecting players for a sports team
- Counting diagonals and triangles in polygons
- “At least” and “at most” selection problems
These question types repeat every year in board exams, making this section extremely important.
Probability Explained
Probability measures how likely an event is to occur. It always lies between 0 and 1.
Basic Probability Formula
P(E) = n(E) / n(S)
Where n(E) is the number of favorable outcomes and n(S) is the total number of possible outcomes in the sample space.
Sample Space and Events
- Sample Space (S): All possible outcomes of an experiment
- Event (E): A specific outcome or set of outcomes
Exercise 7.5 uses everyday examples like coins, dice, and colored balls to explain this concept clearly.
Mutually Exclusive Events and Addition Rule
Two events are mutually exclusive if they cannot happen at the same time. The addition rule depends on whether events overlap:
- P(A∪B) = P(A) + P(B), when A and B are disjoint
- P(A∪B) = P(A) + P(B) − P(A∩B), when A and B overlap
Exercise 7.7 focuses heavily on this rule using cards, dice, and marbles.
Independent Events and Multiplication Rule
Two events are independent if one does not affect the outcome of the other. In such cases:
P(A∩B) = P(A) × P(B)
Exercise 7.8 applies this rule to repeated coin tosses, card draws with replacement, and dice thrown multiple times.
Exercise-Wise Breakdown of Chapter 7
Here is a quick summary covered inside the 11th Class Maths Chapter 7 Notes – Solution:
- Exercise 7.1 – Factorial notation
- Exercise 7.2 – Permutation basics and word problems
- Exercise 7.3 – Repeated letters and circular permutation
- Exercise 7.4 – Combination and its applications
- Exercise 7.5 – Basic probability
- Exercise 7.6 – Experimental probability using frequency tables
- Exercise 7.7 – Addition of probabilities
- Exercise 7.8 – Multiplication of probabilities (independent events)
For the next chapter, check [internal link] for 11th Class Maths Chapter 8 Notes.
Why Download 11th Class Maths Chapter 7 Notes – Solution
Students prefer downloading these solved notes instead of only using the textbook because of the following reasons:
- Every question solved with complete working, not just final answers
- Formulas separated clearly for last-minute revision
- Board-repeated questions marked with board names and years
- Simple explanations for confusing topics like circular permutation
How to Use These Notes for Exam Preparation
- Learn the factorial rule first, since it is used in almost every formula in this chapter.
- Practice permutation and combination questions separately to avoid mixing the two concepts.
- Compare your solved answers with the 11th Class Maths Chapter 7 Notes – Solution to identify mistakes.
- Focus extra time on word problems from Exercise 7.2, 7.3, and 7.4, since these are most common in board papers.
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FAQs
Q1. What topics are included in 11th Class Maths Chapter 7?
Chapter 7 covers factorial notation, permutation, combination, circular permutation, and probability, including addition and multiplication rules for events.
Q2. Where can I find 11th Class Maths Chapter 7 Notes – Solution for free?
You can access the complete solved notes on this page, covering all exercises from 7.1 to 7.8 with step-by-step working for every question.
Q3. What is the difference between permutation and combination?
Permutation considers the order of objects, while combination does not. For example, arranging letters is permutation, but selecting a committee is combination.
Q4. What is the formula for probability?
Probability is calculated as P(E) = n(E)/n(S), where n(E) is the number of favorable outcomes and n(S) is the total number of outcomes in the sample space.
Q5. Is Chapter 7 important for board exams?
Yes, permutation, combination, and probability questions appear almost every year in board papers, especially word problems and probability using dice or cards.
Q6. What is the formula for nCr?
The combination formula is nCr = n! / [r!(n − r)!], used to find the number of ways to select r objects from n objects when order does not matter.