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Fundamental Principle of Counting
If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs: multiply three numbers. Meaning: 'and' multiplies when the choices do not interfere. 'Or' of disjoint options adds (mutually exclusive cases).
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Fundamental Principle of Counting
What is it?
If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs: multiply three numbers. Meaning: 'and' multiplies when the choices do not interfere. 'Or' of disjoint options adds (mutually exclusive cases).
In simple words
- If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices).
- A tree diagram shows the branches.
- Three jobs: multiply three numbers.
- 'And' of independent choices multiplies. Adding 4+3 would be a different question.
Important terms
FUNDAMENTAL PRINCIPLE OF COUNTING: If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs: multiply three numbers. Meaning: 'and' multiplies when the choices do not interfere. 'Or' of disjoint options adds (mutually exclusive cases).
1. PRODUCT RULE: 3 shirts and 2 trousers: 6 outfits. A 4-digit PIN with digits 0-9 and repetition allowed: 10^4. Without repetition: 1098*7 for a 4-digit code from 10 digits (if leading zero allowed — read the question).
2. TREE AND CASES: Draw branches if n is small. Split into cases when a restriction appears (first digit not 0). Add the cases at the end.
What you should learn
- State the main ideas and notation used in fundamental principle of counting.
- Apply the rules of fundamental principle of counting in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- m * n
- Three meals: 322
- Repetition: same n each time
PRODUCT RULE
3 shirts and 2 trousers: 6 outfits. A 4-digit PIN with digits 0-9 and repetition allowed: 10^4. Without repetition: 1098*7 for a 4-digit code from 10 digits (if leading zero allowed — read the question).
TREE AND CASES
Draw branches if n is small. Split into cases when a restriction appears (first digit not 0). Add the cases at the end.
Recommended for Class 11
Full school-level explanation
If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs: multiply three numbers. Meaning: 'and' multiplies when the choices do not interfere. 'Or' of disjoint options adds (mutually exclusive cases).
Multiply paths, then a tree:
WHAT — FUNDAMENTAL PRINCIPLE OF COUNTING
If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs: multiply three numbers. Meaning: 'and' multiplies when the choices do not interfere. 'Or' of disjoint options adds (mutually exclusive cases).
RULE — 1. PRODUCT RULE
3 shirts and 2 trousers: 6 outfits. A 4-digit PIN with digits 0-9 and repetition allowed: 10^4. Without repetition: 1098*7 for a 4-digit code from 10 digits (if leading zero allowed — read the question).
METHOD — 2. TREE AND CASES
Draw branches if n is small. Split into cases when a restriction appears (first digit not 0). Add the cases at the end.
WORKED EXAMPLE
Worked: a meal with 4 mains and 3 drinks. one main AND one drink 4 * 3 = 12 not 4+3
PRODUCT RULE
3 shirts and 2 trousers: 6 outfits. A 4-digit PIN with digits 0-9 and repetition allowed: 10^4. Without repetition: 1098*7 for a 4-digit code from 10 digits (if leading zero allowed — read the question).
Remember
- multiply stages
Examples
- m * n
- Three meals: 322
- Repetition: same n each time
TREE AND CASES
Draw branches if n is small. Split into cases when a restriction appears (first digit not 0). Add the cases at the end.
Remember
- split then add
Examples
- Case even / case odd
- Tree of 2x3=6
- Do not multiply exclusive cases
How to answer a fundamental principle of counting question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- independent 'and' -> multiply
- disjoint 'or' -> add
- 12
- and multiplies
- multiply stages
- split then add
- independent 'and' -> multiply
- disjoint 'or' -> add
How the ideas connect
Worked: a meal with 4 mains and 3 drinks. one main AND one drink 4 * 3 = 12 not 4+3
Deeper understanding
Go beyond memorising Fundamental Principle of Counting. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: a meal with 4 mains and 3 drinks. one main AND one drink 4 * 3 = 12 not 4+3
Common mistakes
Exam focus
Core Idea
1. PRODUCT RULE
2. TREE AND CASES
Worked Example
Exam focus
Quick revision
- independent 'and' -> multiply
- disjoint 'or' -> add
- 12
- and multiplies
- FUNDAMENTAL PRINCIPLE OF COUNTING: If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independe
- 1. PRODUCT RULE: 3 shirts and 2 trousers: 6 outfits. A 4-digit PIN with digits 0-9 and repetition allowed: 10^4. With
'And' of independent choices multiplies. Adding 4+3 would be a different question.
For long answers on Fundamental Principle of Counting: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- independent 'and' -> multiply
- disjoint 'or' -> add
- 12
- and multiplies
Important formulas
- independent 'and' -> multiply
- disjoint 'or' -> add
- 12
Common exam traps
- 'And' of independent choices multiplies. Adding 4+3 would be a different question.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
'And' of independent choices multiplies. Adding 4+3 would be a different question.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
'And' of independent choices multiplies. Adding 4+3 would be a different question.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
independent 'and' -> multiply
disjoint 'or' -> add
12
and multiplies
If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs:
- Definition
- If job A can be done in m ways and job B in n ways, then A then B can be done in m*n ways (independent choices). A tree diagram shows the branches. Three jobs:
- Example
- m * n
- Critical warning
- 'And' of independent choices multiplies. Adding 4+3 would be a different question.
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Exam Practice
Define
Q1. Define Fundamental Principle of Counting.
Explain
Q2. Explain: PRODUCT RULE.
Example
Q3. Give one clear example of Fundamental Principle of Counting and say why it fits.
Mistake
Q4. State one common mistake students make about Fundamental Principle of Counting.
Exam note
Q5. Write a short exam note on Fundamental Principle of Counting: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.