Kenneth ChengMathematical modeling
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Permutations and combinations

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

高三選擇性必修 第三册(A版).pdf · 6.2 · PDF 19 / printed page 14

Revisit first: Addition and multiplication counting principles

TOPIC 01

Permutations and combinations

Distinguish ordered selections, unordered sets and symmetry in arrangements.

What you will be able to explain

  • Distinguish ordered selections, unordered sets and symmetry in arrangements.
  • Justify the method and check the conditions in a new situation.

Defining relation

Distinguish ordered selections, unordered sets and symmetry in arrangements.

Pnr=n!/(n−r)!;Cnr=n!/[r!(n−r)!]P_n^r=n!/(n-r)!;\quad C_n^r=n!/[r!(n-r)!]

Conditions

Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Why does choosing a committee differ from choosing president and secretary?

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: Compare labeled assignments and identical objects in a small exhaustive list.

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 a president and secretary from m distinct people. Count assignments.

m=7m=7
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
Pm2=m(m−1)P_m^2=m(m-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.

    N=7(7−1)=42N=7(7-1)=42

  3. The two offices are different, so order matters.

The requested value is 42.

Checks and common pitfalls: The two offices are different, so order matters.

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

02 / Standard#Worked example

From m people, form a 3-person committee that includes a specified person.

m=8m=8
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
Cm−12C_{m-1}^2
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=C72=21N=C_{7}^2=21

  3. Select only the remaining two members from the other people.

The requested value is 21.

Checks and common pitfalls: Select only the remaining two members from the other people.

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

03 / Transfer#Worked example

Arrange m distinct people in a line with A and B adjacent.

m=9m=9
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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
Treat A,B as one block.
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(m−1)!=80640N=2(m-1)!=80640

  3. Multiply by two for AB and BA inside the block.

The requested value is 80640.

Checks and common pitfalls: Multiply by two for AB and BA inside the block.

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

04 / Foundation#Your turn

Choose a president and secretary from m distinct people. Count assignments.

m=10m=10
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
Pm2=m(m−1)P_m^2=m(m-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.

    N=10(10−1)=90N=10(10-1)=90

  3. The two offices are different, so order matters.

The requested value is 90.

Checks and common pitfalls: The two offices are different, so order matters.

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

05 / Foundation#Your turn

Choose an unordered committee of 3 from m people.

m=11m=11
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
Cm3=Pm3/3!C_m^3=P_m^3/3!
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=11(10)(9)/6=165N=11(10)(9)/6=165

  3. Each committee was counted in six possible orders.

The requested value is 165.

Checks and common pitfalls: Each committee was counted in six possible orders.

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

06 / Foundation#Your turn

Arrange m distinct people in a circle, considering rotations identical but reflections different.

m=12m=12
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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=(m−1)!N=(m-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.

    N=39916800N=39916800

  3. Fix one person to remove rotational duplicates.

The requested value is 39916800.

Checks and common pitfalls: Fix one person to remove rotational duplicates.

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

07 / Foundation#Your turn

Count arrangements containing r identical As, two identical Bs and one C.

r=2r=2
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
(r+3)!/(r!2!)(r+3)!/(r!2!)
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!/(2!2!)=30N=5!/(2!2!)=30

  3. Permutations of each identical-letter group do not create new arrangements.

The requested value is 30.

Checks and common pitfalls: Permutations of each identical-letter group do not create new arrangements.

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

08 / Standard#Your turn

Count arrangements containing r identical As, two identical Bs and one C.

r=3r=3
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
(r+3)!/(r!2!)(r+3)!/(r!2!)
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!/(3!2!)=60N=6!/(3!2!)=60

  3. Permutations of each identical-letter group do not create new arrangements.

The requested value is 60.

Checks and common pitfalls: Permutations of each identical-letter group do not create new arrangements.

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

09 / Standard#Your turn

From m people, form a 3-person committee that includes a specified person.

m=15m=15
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
Cm−12C_{m-1}^2
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=C142=91N=C_{14}^2=91

  3. Select only the remaining two members from the other people.

The requested value is 91.

Checks and common pitfalls: Select only the remaining two members from the other people.

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

10 / Standard#Your turn

Place t identical balls in 3 labeled boxes; empty boxes are allowed. Count distributions.

t=11t=11
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
x1+x2+x3=t,xi≥0x_1+x_2+x_3=t,\quad x_i\ge0
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=C132=78N=C_{13}^2=78

  3. Stars and bars count integer allocations; labeled boxes remain distinct.

The requested value is 78.

Checks and common pitfalls: Stars and bars count integer allocations; labeled boxes remain distinct.

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

11 / Standard#Your turn

Arrange m distinct people in a line with A and B adjacent.

m=6m=6
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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
Treat A,B as one block.
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(m−1)!=240N=2(m-1)!=240

  3. Multiply by two for AB and BA inside the block.

The requested value is 240.

Checks and common pitfalls: Multiply by two for AB and BA inside the block.

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

12 / Transfer#Your turn

Place t identical balls in 3 labeled boxes; empty boxes are allowed. Count distributions.

t=13t=13
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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.
x1+x2+x3=t,xi≥0x_1+x_2+x_3=t,\quad x_i\ge0
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=C152=105N=C_{15}^2=105

  3. Stars and bars count integer allocations; labeled boxes remain distinct.

The requested value is 105.

Checks and common pitfalls: Stars and bars count integer allocations; labeled boxes remain distinct.

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

13 / Transfer#Your turn

Arrange m distinct people in a line with A and B adjacent.

m=8m=8
  • Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.
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
Treat A,B as one block.
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(m−1)!=10080N=2(m-1)!=10080

  3. Multiply by two for AB and BA inside the block.

The requested value is 10080.

Checks and common pitfalls: Multiply by two for AB and BA inside the block.

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

    • Distinguish ordered selections, unordered sets and symmetry in arrangements.
    • Which condition is essential in permutations and combinations?
    • Why does choosing a committee differ from choosing president and secretary?

    Board plan

    • Defining relation: Distinguish ordered selections, unordered sets and symmetry in arrangements.
      Pnr=n!/(n−r)!;Cnr=n!/[r!(n−r)!]P_n^r=n!/(n-r)!;\quad C_n^r=n!/[r!(n-r)!]
    • Conditions: Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.

    Anticipated thinking

    • Dividing by r! is valid only when every selected set has exactly r! counted orders.

    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 ↗