Chapter 07 – Factorial, Permutations and Combinations

 

Chapter 07 – Factorial, Permutations and Combinations

Chapter 07 of First Year Mathematics focuses on one of the most important and practical areas of mathematics: Factorial, Permutations, and Combinations. This chapter builds the foundation for solving counting problems, probability, and real-life arrangements. It is widely used in exams, competitive tests, and everyday problem-solving situations.

In this chapter, students learn how to count objects efficiently without listing all possibilities. The concepts may look simple at first, but they require clear understanding and practice.

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Introduction to Factorial

The concept of factorial is the base of permutations and combinations. It is represented by the symbol “!”.

Definition:

For any positive integer n,

n! = n × (n − 1) × (n − 2) × ... × 2 × 1

Examples:

5! = 5 × 4 × 3 × 2 × 1 = 120
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1 = 6

Special Case:

0! = 1

This is a very important rule in mathematics.

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Basic Properties of Factorial

1. n! = n × (n − 1)!
2. 1! = 1
3. 0! = 1
4. n! grows very fast as n increases

Example:

6! = 6 × 5!
= 6 × 120
= 720

Factorials are used in counting arrangements and selections.

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Introduction to Permutations

Permutation means arrangement of objects in a specific order.

👉 Order matters in permutations.

Formula:

nPr = n! / (n − r)!

Where:
n = total number of objects
r = number of objects to arrange

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Examples of Permutations

Example 1:

Find 5P2

5P2 = 5! / (5 − 2)!
= 5! / 3!
= (5 × 4 × 3!) / 3!
= 5 × 4
= 20

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Example 2:

How many ways can 3 students stand from 6 students?

6P3 = 6! / 3!
= 6 × 5 × 4
= 120 ways

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Types of Permutations

1. Permutation without repetition

nPr = n! / (n − r)!

2. Permutation with repetition

Number of ways = n^r

Example:
If 3 letters are used to form 2-letter words:

3^2 = 9

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3. Circular Permutation

Formula:

(n − 1)!

Example:
4 people sitting around a table:

(4 − 1)! = 3! = 6 ways

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Introduction to Combinations

Combination means selection of objects where order does NOT matter.

👉 Order does NOT matter in combinations.

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Formula of Combination

nCr = n! / [r! (n − r)!]

Where:
n = total objects
r = objects selected

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Examples of Combinations

Example 1:

Find 5C2

5C2 = 5! / [2! 3!]
= (5 × 4 × 3!) / (2 × 1 × 3!)
= (5 × 4) / 2
= 10

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Example 2:

Select 3 students from 6:

6C3 = 6! / (3! 3!)
= 20 ways

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Important Properties of Combination

1. nC0 = 1
2. nCn = 1
3. nCr = nC(n − r)

Example:

5C2 = 5C3 = 10

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Difference Between Permutation and Combination

Concept	Meaning	Order
Permutation	Arrangement	Order matters
Combination	Selection	Order does not matter

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Real Life Applications

Factorial, permutations, and combinations are used in many real-life situations:

• Arranging books on a shelf
• Seating people in a row or circle
• Forming teams from players
• Password and code generation
• Lottery and probability problems

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Word Problems Understanding

This chapter also teaches how to solve real-life problems:

👉 If order matters → use permutation
👉 If order does not matter → use combination

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Example Problem:

From 5 people, how many committees of 2 can be formed?

Solution:
5C2 = 10

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Example Problem:

How many ways to arrange 4 books?

Solution:
4! = 24

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PDF SOLUTION:

Tips for Students

• Always identify whether the question is permutation or combination
• Practice factorial simplification
• Learn formulas by heart
• Solve MCQs for better understanding
• Avoid mistakes in cancellation

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Common Mistakes

• Confusing permutation with combination
• Forgetting factorial rules
• Incorrect simplification
• Ignoring 0! = 1

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Practice Importance

This chapter requires regular practice. Students should solve:

• Exercise questions
• MCQs
• Past papers

This will help in building strong concepts and improving speed.

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Conclusion

Chapter 07 – Factorial, Permutations, and Combinations is a very important chapter in mathematics. It develops logical thinking and problem-solving skills. Understanding this chapter makes it easier to study probability and advanced mathematics in the future.

Students must focus on formulas, concepts, and practice regularly to master this chapter. With proper understanding, this topic becomes easy and interesting.