Kenneth ChengMathematical modeling
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Addition and multiplication counting principles

Read the idea, work independently, then explain what changed.

高三選擇性必修 第三册(A版).pdf · 6.1 · PDF 7 / printed page 2

TOPIC 01

Addition and multiplication counting principles

Count disjoint alternatives and sequential choices without overlap or omission.

What you will be able to explain

  • Count disjoint alternatives and sequential choices without overlap or omission.
  • Justify the method and check the conditions in a new situation.

Defining relation

Count disjoint alternatives and sequential choices without overlap or omission.

N=∑NiorN=∏NiN=\sum N_i\quad\text{or}\quad N=\prod N_i

Conditions

Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: When should two counts be added rather than multiplied?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Choosing 2 distinct objects from 5: 10 unordered choices, 20 ordered arrangements. If r>n both counts are zero; choosing none gives one empty choice.C(5,2) = 10P(5,2) = 20

Choosing 2 distinct objects from 5: 10 unordered choices, 20 ordered arrangements. If r>n both counts are zero; choosing none gives one empty choice.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Draw a branching tree with unequal numbers of second-stage choices.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Choose one book: t mathematics titles or 3 history titles, with no overlap. How many choices?

t=2t=2
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=t+3N=t+3
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=2+3=5N=2+3=5

  3. The two subject cases are disjoint alternatives.

The requested value is 5.

Checks and common pitfalls: The two subject cases are disjoint alternatives.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

There are t first-stage choices with 2 continuations each, and 3 other choices with 5 continuations each. Count paths.

t=3t=3
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=2t+3⋅5N=2t+3\cdot5
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=6+15=21N=6+15=21

  3. Unequal branch sizes require addition of branch products.

The requested value is 21.

Checks and common pitfalls: Unequal branch sizes require addition of branch products.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Use digits 0,…,t−1 to form two-digit numbers without repeated digits, where 2≤t≤9. Count them.

t=6t=6
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
(t−1)(t−1)(t-1)(t-1)
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    first choices=5\text{first choices}=5

  3. Apply the stated relation and retain its conditions.

    second choices=5\text{second choices}=5

  4. Apply the stated relation and retain its conditions.

    N=25N=25

  5. The first digit cannot be zero; the second may be zero but cannot repeat the first.

The requested value is 25.

Checks and common pitfalls: The first digit cannot be zero; the second may be zero but cannot repeat the first.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Choose one book: t mathematics titles or 3 history titles, with no overlap. How many choices?

t=5t=5
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=t+3N=t+3
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=5+3=8N=5+3=8

  3. The two subject cases are disjoint alternatives.

The requested value is 8.

Checks and common pitfalls: The two subject cases are disjoint alternatives.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Choose one of t shirts and one of 4 trousers. How many outfits?

t=6t=6
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=t⋅4N=t\cdot4
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=24N=24

  3. Every shirt permits all four trouser choices.

The requested value is 24.

Checks and common pitfalls: Every shirt permits all four trouser choices.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A code is one of t letters followed by a digit 0–9. How many codes?

t=7t=7
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=t⋅10N=t\cdot10
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=70N=70

  3. A leading-zero restriction does not apply to the second symbol in this code.

The requested value is 70.

Checks and common pitfalls: A leading-zero restriction does not apply to the second symbol in this code.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Among t+8 students, t study art, 8 study music and 3 study both. How many study either subject?

t=8t=8
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=8+8−3=13N=8+8-3=13

  3. Subtract the overlap once because it was counted twice.

The requested value is 13.

Checks and common pitfalls: Subtract the overlap once because it was counted twice.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Among t+8 students, t study art, 8 study music and 3 study both. How many study either subject?

t=9t=9
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=9+8−3=14N=9+8-3=14

  3. Subtract the overlap once because it was counted twice.

The requested value is 14.

Checks and common pitfalls: Subtract the overlap once because it was counted twice.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

There are t first-stage choices with 2 continuations each, and 3 other choices with 5 continuations each. Count paths.

t=10t=10
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=2t+3⋅5N=2t+3\cdot5
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=20+15=35N=20+15=35

  3. Unequal branch sizes require addition of branch products.

The requested value is 35.

Checks and common pitfalls: Unequal branch sizes require addition of branch products.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Passwords of length 3 use t symbols with repetition allowed. Exclude those with all symbols identical. Count remaining passwords.

t=11t=11
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=t3−tN=t^3-t
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=1331−11=1320N=1331-11=1320

  3. The excluded all-identical passwords are one per symbol, not one total.

The requested value is 1320.

Checks and common pitfalls: The excluded all-identical passwords are one per symbol, not one total.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Use digits 0,…,t−1 to form two-digit numbers without repeated digits, where 2≤t≤9. Count them.

t=7t=7
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
(t−1)(t−1)(t-1)(t-1)
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    first choices=6\text{first choices}=6

  3. Apply the stated relation and retain its conditions.

    second choices=6\text{second choices}=6

  4. Apply the stated relation and retain its conditions.

    N=36N=36

  5. The first digit cannot be zero; the second may be zero but cannot repeat the first.

The requested value is 36.

Checks and common pitfalls: The first digit cannot be zero; the second may be zero but cannot repeat the first.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Passwords of length 3 use t symbols with repetition allowed. Exclude those with all symbols identical. Count remaining passwords.

t=13t=13
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
N=t3−tN=t^3-t
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    N=2197−13=2184N=2197-13=2184

  3. The excluded all-identical passwords are one per symbol, not one total.

The requested value is 2184.

Checks and common pitfalls: The excluded all-identical passwords are one per symbol, not one total.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Use digits 0,…,t−1 to form two-digit numbers without repeated digits, where 2≤t≤9. Count them.

t=2t=2
  • Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Decide whether choices are alternative cases or successive stages, then check overlap.
Hint 2
Use this intermediate relation.
(t−1)(t−1)(t-1)(t-1)
Worked solution

  1. Decide whether choices are alternative cases or successive stages, then check overlap.

  2. Apply the stated relation and retain its conditions.

    first choices=1\text{first choices}=1

  3. Apply the stated relation and retain its conditions.

    second choices=1\text{second choices}=1

  4. Apply the stated relation and retain its conditions.

    N=1N=1

  5. The first digit cannot be zero; the second may be zero but cannot repeat the first.

The requested value is 1.

Checks and common pitfalls: The first digit cannot be zero; the second may be zero but cannot repeat the first.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Count disjoint alternatives and sequential choices without overlap or omission.
    • Which condition is essential in addition and multiplication counting principles?
    • When should two counts be added rather than multiplied?

    Board plan

    • Defining relation: Count disjoint alternatives and sequential choices without overlap or omission.
      N=∑NiorN=∏NiN=\sum N_i\quad\text{or}\quad N=\prod N_i
    • Conditions: Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.

    Anticipated thinking

    • Having two choices to describe does not automatically mean multiply.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗