Exercise 7.2 | Permutations | New 11th Class Math 2025| New 11th Class Math 2025
Introduction:
In the new First Year Mathematics (2025 Edition) syllabus, Chapter 7 deals with Permutations and Combinations. Exercise 7.2 focuses specifically on permutations, which is one of the most important concepts in probability, statistics, and combinatorics.
A permutation means arranging things in a particular order. Whenever order matters, we use permutations. For example, if you have 3 students A, B, and C, and you want to arrange them in a line, the possible arrangements are:
ABC, ACB, BAC, BCA, CAB, CBA
Here, each arrangement is different because the order has changed. This is what makes permutations different from combinations, where order does not matter.
In this exercise, you will learn formulas, properties, and applications of permutations. It will prepare you for higher-level problems in probability and real-life applications like seating arrangements, digits in numbers, and arrangements of objects.
Concept of Factorial (n!)
Before learning permutations, you need to understand the factorial function.
For a natural number n,
n! = n × (n - 1) × (n - 2) × … × 3 × 2 × 1
For example:
5! = 5 × 4 × 3 × 2 × 1 = 120
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1 = 6
1! = 1
0! = 1 (by definition)
The factorial function helps us count how many ways we can arrange objects.
Definition of Permutation
A permutation of a set of objects is any arrangement of those objects in a certain order.
If n different objects are available, and we want to select r of them and arrange them, the number of permutations is written as:
P(n, r) = n! / (n - r)!
Where:
n = total number of objects
r = number of objects chosen at a time
This formula only works when order matters.
Example 1:
How many ways can 3 students be arranged in a row?
Here, n = 3, r = 3
P(3, 3) = 3! / (3 - 3)! = 3! / 0! = 6 / 1 = 6
So, there are 6 arrangements:
ABC, ACB, BAC, BCA, CAB, CBA
Example 2:
How many 2-digit numbers can be formed using digits {1, 2, 3, 4} without repetition?
Here, n = 4, r = 2
P(4, 2) = 4! / (4 - 2)! = 4 × 3 = 12
So, 12 different 2-digit numbers can be formed.
Important Cases
Case 1: When all n objects are arranged
If r = n, then:
P(n, n) = n!
For example, arranging 5 books on a shelf:
P(5, 5) = 5! = 120 ways
Case 2: When only r objects are chosen
If n = 6 and r = 2:
P(6, 2) = 6 × 5 = 30 ways
Formula Recap
General formula:
P(n, r) = n! / (n - r)!
Special case:
P(n, n) = n!
Difference Between Permutations and Combinations
Aspect Permutation Combination
Order matters? Yes No
Formula n! / (n - r)! n! / [r!(n-r)!]
Example (A, B, C with r = 2) AB, BA, AC, CA, BC, CB AB, AC, BC
Application of Permutations
Permutations are not only theoretical; they are widely used in real life. Some applications include:
1. Seating arrangements: Arranging guests around a table.
2. Passwords and codes: Counting possible arrangements of letters/digits.
3. Sports tournaments: Deciding the order of players in matches.
4. Telephone numbers: Counting unique number arrangements.
5. Genetics and biology: Arranging DNA sequences.
Solved Problems
Problem 1:
Find the number of ways to arrange 5 students in a row.
Solution:
n = 5, r = 5
P(5, 5) = 5! = 120
So, there are 120 ways.
Problem 2:
Find the number of ways to arrange 3 letters out of 6 letters.
Solution:
n = 6, r = 3
P(6, 3) = 6! / (6 - 3)! = 6 × 5 × 4 = 120
So, 120 ways.
PDF SOLUTION:
Problem 3:
How many 4-digit numbers can be formed from digits {1, 2, 3, 4, 5} without repetition?
Solution:
n = 5, r = 4
P(5, 4) = 5! / (5 - 4)! = 5 × 4 × 3 × 2 = 120
So, 120 numbers.
Special Note on Repetition
In many real problems, repetition is allowed. For example, forming passwords where letters can be repeated.
If repetition is allowed:
Number of arrangements = n^r
Example: If digits {1, 2, 3} are used to form 2-digit numbers with repetition allowed:
3^2 = 9 numbers.
These are: 11, 12, 13, 21, 22, 23, 31, 32, 33
1. Identify whether the problem is permutation or combination.
(If order matters → permutation).
2. Find values of n and r.
3. Apply the formula P(n, r) = n! / (n - r)!
4. Simplify factorials to get the result.
5. Check for repetition (allowed or not).
Summary
Permutation means arrangement where order matters.
Formula: P(n, r) = n! / (n - r)!
If r = n, then permutations = n!
With repetition allowed: n^r
Permutations are widely used in real-life problems like arrangements, numbers, and passwords.
Conclusion
Exercise 7.2 (Permutations) in the New 11th Class Math 2025 syllabus is one of the most practical and useful topics in mathematics. It develops logical thinking and problem-solving skills. Mastering this exercise will help you not only in exams but also in advanced topics like probability, statistics, and computer science.
By practicing different problems, you will understand how permutations change when the number of objects increases, when repetition is allowed, or when restrictions are applied. This exercise builds a strong base for Combinations (Exercise 7.3) and later chapters in probability.
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