Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Binomial theorem

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

高三選擇性必修 第三册(A版).pdf · 6.3 · PDF 34 / printed page 29

Revisit first: Permutations and combinations

TOPIC 01

Binomial theorem

Use the general term to locate coefficients, constants and coefficient sums.

What you will be able to explain

  • Use the general term to locate coefficients, constants and coefficient sums.
  • Justify the method and check the conditions in a new situation.

Defining relation

Use the general term to locate coefficients, constants and coefficient sums.

(a+b)n=∑r=0nCnran−rbr(a+b)^n=\sum_{r=0}^nC_n^ra^{n-r}b^r

Conditions

The exponent n is a nonnegative integer; term number is r+1, not r.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does a negative second term change all coefficients to negative?

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: Use parity of the selection index to explain the sign pattern.

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

Find the coefficient of x in (1+x)^m.

m=6m=6
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
Cm1=mC_m^1=m
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    [x](1+x)6=6[x](1+x)^{6}=6

  3. Selecting x from exactly one factor gives the coefficient.

The requested value is 6.

Checks and common pitfalls: Selecting x from exactly one factor gives the coefficient.

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

02 / Standard#Worked example

Find the constant term in (x+1/x)^2m.

m=7m=7
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
x2m−rx−r=x2m−2rx^{2m-r}x^{-r}=x^{2m-2r}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    14−2r=0⇒r=714-2r=0\Rightarrow r=7

  3. Apply the stated relation and retain its conditions.

    T=C147=3432T=C_{14}^{7}=3432

  4. A zero exponent locates the constant term; then evaluate its coefficient.

The requested value is 3432.

Checks and common pitfalls: A zero exponent locates the constant term; then evaluate its coefficient.

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

03 / Transfer#Worked example

Find the remainder when (1+10)^m is divided by 100.

m=8m=8
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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
Terms with r≥2 contain 100.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    118≡1+10(8)(mod100)11^{8}\equiv1+10(8)\pmod{100}

  3. Apply the stated relation and retain its conditions.

    R=81R=81

  4. Reduce the surviving expression modulo 100; it may still exceed 99.

The requested value is 81.

Checks and common pitfalls: Reduce the surviving expression modulo 100; it may still exceed 99.

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

04 / Foundation#Your turn

Find the coefficient of x in (1+x)^m.

m=9m=9
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
Cm1=mC_m^1=m
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    [x](1+x)9=9[x](1+x)^{9}=9

  3. Selecting x from exactly one factor gives the coefficient.

The requested value is 9.

Checks and common pitfalls: Selecting x from exactly one factor gives the coefficient.

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

05 / Foundation#Your turn

Find the coefficient of x² in (1+x)^m.

m=10m=10
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
Cm2=m(m−1)/2C_m^2=m(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.

    [x2]=45[x^2]=45

  3. The unordered pair of chosen factors determines the term.

The requested value is 45.

Checks and common pitfalls: The unordered pair of chosen factors determines the term.

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

06 / Foundation#Your turn

Find the sum of all coefficients of (2+x)^m.

m=11m=11
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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
Substitute x=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.

    (2+1)11=177147(2+1)^{11}=177147

  3. Use x=1 for the full coefficient sum, not 2^m unless both base coefficients equal one.

The requested value is 177147.

Checks and common pitfalls: Use x=1 for the full coefficient sum, not 2^m unless both base coefficients equal one.

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

07 / Foundation#Your turn

Find the coefficient of x² in (1−2x)^m.

m=12m=12
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
Cm2(−2)2C_m^2(-2)^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.

    [x2]=4C122=264[x^2]=4C_{12}^2=264

  3. An even selection index makes this coefficient positive.

The requested value is 264.

Checks and common pitfalls: An even selection index makes this coefficient positive.

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

08 / Standard#Your turn

Find the coefficient of x² in (1−2x)^m.

m=13m=13
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
Cm2(−2)2C_m^2(-2)^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.

    [x2]=4C132=312[x^2]=4C_{13}^2=312

  3. An even selection index makes this coefficient positive.

The requested value is 312.

Checks and common pitfalls: An even selection index makes this coefficient positive.

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

09 / Standard#Your turn

Find the constant term in (x+1/x)^2m.

m=14m=14
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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.
x2m−rx−r=x2m−2rx^{2m-r}x^{-r}=x^{2m-2r}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    28−2r=0⇒r=1428-2r=0\Rightarrow r=14

  3. Apply the stated relation and retain its conditions.

    T=C2814=40116600T=C_{28}^{14}=40116600

  4. A zero exponent locates the constant term; then evaluate its coefficient.

The requested value is 40116600.

Checks and common pitfalls: A zero exponent locates the constant term; then evaluate its coefficient.

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

10 / Standard#Your turn

Find the alternating sum of binomial coefficients for m≥1.

∑r=04(−1)rC4r\sum_{r=0}^{4}(-1)^rC_{4}^r
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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
Substitute x=-1 in (1+x)^m.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    (1−1)4=0(1-1)^{4}=0

  3. The exponent is positive, so the substitution is unambiguous.

The requested value is 0.

Checks and common pitfalls: The exponent is positive, so the substitution is unambiguous.

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

11 / Standard#Your turn

Find the remainder when (1+10)^m is divided by 100.

m=5m=5
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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
Terms with r≥2 contain 100.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    115≡1+10(5)(mod100)11^{5}\equiv1+10(5)\pmod{100}

  3. Apply the stated relation and retain its conditions.

    R=51R=51

  4. Reduce the surviving expression modulo 100; it may still exceed 99.

The requested value is 51.

Checks and common pitfalls: Reduce the surviving expression modulo 100; it may still exceed 99.

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

12 / Transfer#Your turn

Find the alternating sum of binomial coefficients for m≥1.

∑r=06(−1)rC6r\sum_{r=0}^{6}(-1)^rC_{6}^r
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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
Substitute x=-1 in (1+x)^m.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    (1−1)6=0(1-1)^{6}=0

  3. The exponent is positive, so the substitution is unambiguous.

The requested value is 0.

Checks and common pitfalls: The exponent is positive, so the substitution is unambiguous.

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

13 / Transfer#Your turn

Find the remainder when (1+10)^m is divided by 100.

m=7m=7
  • The exponent n is a nonnegative integer; term number is r+1, not r.
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
Terms with r≥2 contain 100.
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    117≡1+10(7)(mod100)11^{7}\equiv1+10(7)\pmod{100}

  3. Apply the stated relation and retain its conditions.

    R=71R=71

  4. Reduce the surviving expression modulo 100; it may still exceed 99.

The requested value is 71.

Checks and common pitfalls: Reduce the surviving expression modulo 100; it may still exceed 99.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • Use the general term to locate coefficients, constants and coefficient sums.
    • Which condition is essential in binomial theorem?
    • Does a negative second term change all coefficients to negative?

    Board plan

    • Defining relation: Use the general term to locate coefficients, constants and coefficient sums.
      (a+b)n=∑r=0nCnran−rbr(a+b)^n=\sum_{r=0}^nC_n^ra^{n-r}b^r
    • Conditions: The exponent n is a nonnegative integer; term number is r+1, not r.

    Anticipated thinking

    • A binomial coefficient and the full coefficient of a term may differ.

    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.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗