1st Year Math Chapter 7 Exercise 7.1 Notes: Fundamental Principle of Counting (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Chapter 7, Permutations and Combinations, follows Sequences and Series in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 7.1 opens the chapter with the basic counting principles and factorial notation that permutations and combinations are built on.
What Does This Exercise Cover?
This exercise teaches the two fundamental rules for counting the number of ways multiple choices or tasks can occur together, along with factorial notation, which becomes essential from Exercise 7.2 onward.
Key Concepts
1. The Fundamental (Multiplication) Principle of Counting
If one task can be completed in m ways, and a second, independent task can be completed in n ways, then both tasks together (one after the other) can be completed in m × n ways.
2. The Addition Principle of Counting
If a task can be completed in m ways OR in n ways, but not both at once (mutually exclusive options), then the task can be completed in m + n ways.
3. Factorial Notation
For a positive integer n, n! (read ‘n factorial’) means n × (n−1) × (n−2) × … × 2 × 1. By definition, 0! = 1.
Common Factorial Values
| n | n! |
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
Solved Examples
Example 1: A man has 3 shirts and 4 pairs of trousers. In how many ways can he get dressed?
Choosing a shirt: 3 ways. Choosing trousers: 4 ways.
By the multiplication principle: 3 × 4.
Answer: 12 ways
Example 2: A code consists of 2 letters (from 26) followed by 3 digits (from 10), with repetition allowed. How many codes are possible?
Letters: 26 × 26. Digits: 10 × 10 × 10.
Total: 26 × 26 × 10 × 10 × 10.
Answer: 676,000 codes
Example 3: Evaluate 6! / (4! × 2!).
6! = 720, 4! = 24, 2! = 2.
720 / (24 × 2) = 720 / 48.
Answer: 15
Sample MCQs
1. If one task can be done in m ways and another (independent) task in n ways, both together can be done in:
a) m + n ways b) m × n ways c) m − n ways d) m / n ways
Answer: b) m × n ways
2. The value of 0! is:
a) 0 b) 1 c) Undefined d) −1
Answer: b) 1
3. The value of 5! is:
a) 60 b) 100 c) 120 d) 24
Answer: c) 120
4. If a task can be done in m ways OR n ways (not both), the total number of ways is:
a) m × n b) m + n c) m − n d) mn / 2
Answer: b) m + n
5. 4! equals:
a) 12 b) 16 c) 24 d) 4
Answer: c) 24
Important Short Questions
- State the fundamental (multiplication) principle of counting.
- State the addition principle of counting.
- Define n! (n factorial).
- Evaluate 7! / 5!.
- A man has 4 shirts and 3 ties. In how many ways can he choose one shirt and one tie?
Important Long Questions
- A license plate consists of 3 letters followed by 4 digits. How many different license plates are possible if repetition is allowed?
- Evaluate 8! / (3! × 5!).
- Explain the difference between the multiplication principle and the addition principle of counting, giving an example of each.
- A restaurant menu offers 5 starters, 6 main courses, and 4 desserts. In how many ways can a customer choose one item from each course?
How to Approach This Exercise Effectively
- Ask yourself ‘AND’ or ‘OR’ for every counting problem: ‘AND’ signals multiplication, ‘OR’ (mutually exclusive) signals addition.
- Practice simplifying factorial expressions by cancelling matching terms in the numerator and denominator before multiplying anything out.
- Memorize factorial values up to at least 6! or 7! — recalculating them from scratch under exam pressure wastes time.
- For multi-step problems (like license plates), break the problem into separate independent choices before multiplying.
- Double-check whether repetition is allowed in a problem — it changes whether you multiply the same number repeatedly or count down each time.
FAQs
Q: What is the difference between the addition and multiplication principles?
A: Use multiplication when tasks happen together (AND), and addition when only one of several mutually exclusive options happens (OR).
Q: Why is 0! defined as 1?
A: It’s a mathematical convention that keeps factorial-based formulas (like permutations and combinations) consistent and well-defined even when zero items are chosen.
Q: Can the multiplication principle be extended to more than two tasks?
A: Yes — it extends to any number of independent tasks; simply multiply the number of ways for each one together.
Q: What real-world situations use the fundamental principle of counting?
A: Examples include counting possible passwords, license plates, outfit combinations, or menu choices — anywhere multiple independent choices combine.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 7: Permutations and Combinations, Exercise 7.1.