MathematicsChapter 6: Permutation and Combination
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).
Learning objectives
- 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.
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: multiply three numbers. Meaning: 'and' multiplies when the choices do not interfere. 'Or' of disjoint options adds (mutually exclusive cases).
Where / symbols
- 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: 10*9*8*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.
Overview
Multiply paths, then a tree:
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).
Examples m * n · Three meals: 3*2*2 · 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.
Examples Case even / case odd · Tree of 2x3=6 · Do not multiply exclusive cases
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
How to answer a fundamental principle of counting question
- 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.
Worked example
- a meal with 4 mains and 3 drinks.
- one main AND one drink
- 4 * 3 = 12
- not 4+3
In simple words
'And' of independent choices multiplies. Adding 4+3 would be a different question.
Key points
- independent 'and' -> multiply
- disjoint 'or' -> add
- multiply stages
- split then add
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Fundamental Principle of Counting
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
Worked Example
- Worked: a meal with 4 mains and 3 drinks.
- one main AND one drink
- 4 * 3 = 12
- not 4+3
Common Misconception
- 'And' of independent choices multiplies. Adding 4+3 would be a different question.
FBISE Exam Check
- State the rule for fundamental principle of counting before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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