What is seven choose zero?

1 There is exactly one selection containing nothing.

Expand a plus b raised to five, with a's powers decreasing. What comes third?

10a^3b^2 The coefficient multiplies both powers; the b part is not added separately.

Here x and y name two quantities. What term ends x minus y raised to six?

+y^6 Six negative factors multiply to give a positive coefficient.

What does twice x, all cubed, become?

8x^3 The factor 2 is multiplied three times, just like x.

Which split of 96 uses a power of ten with the smallest whole-number adjustment?

100 minus 4 Powers of 100 are easy, and subtracting 4 gives the smallest whole-number adjustment.

Does 2 overestimate or underestimate this power?

Underestimate. The omitted terms are positive and increase the total.

Take three raised to n, minus twice n. What smallest positive whole number leaves a multiple of 4 when subtracted?

1 Removing 1 leaves only an integer multiple of 4.

How far above the exact value is this three-term approximation?

0.000001 The omitted term is negative, so leaving it out raises the result.

Teach Binomial Theorem in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Repeated percentage increases multiply because each increase uses the latest price.
Growing a square changes its area through both its length and its width.
Packing items into equal groups can leave leftovers that remainders describe.

What to teach

How to teach it

Build the expansion

Pascal’s triangle linked to choice counts and a fourth-power expansion with one coefficient missing.

1. Pascal's triangle and binomial coefficients

Neighbouring entries add to give coefficients for the next power.

Show: In the diagram, a and b name quantities. Follow the arrows from the threes. Match the bottom row to the expansion and its choice counts.
Ask: What coefficient fills the gap without multiplying the four brackets?
Say: The missing coefficient is 6, from adding the two threes above. The row supplies exact coefficients without multiplying every bracket. Read the small symbols as 'four choose zero' through 'four choose four'.
Used in: Counting snack selections avoids repeating choices.

If your child says…“Choosing nothing gives zero ways.”

Ask: How many ways can you choose a from every bracket?
What’s true: Choosing no b means choosing a from every bracket: exactly one choice.

Bracket selections explain coefficients, the general expansion, and the induction step.

2. Binomial theorem for positive integral indices

Count letter selections to find each product’s coefficient.

Show: Follow the diagram from selections to products, then the general rule and induction arrow.
Ask: What coefficient collects selections containing exactly one b?
Say: Three selections give three a squared b. Opposing powers give n plus one terms. From power one, adding brackets combines neighbouring coefficients, proving the rule by induction.
Used in: Repeated percentage growth calculations track savings.

If your child says…“I multiply the coefficient by the a part, then add the b part.”

Ask: In each mixed product, are the a and b factors multiplied or added?
What’s true: Each term multiplies its choice count by both powers.

$(1-x)^4=(1+(-x))^4$

$=1+4(-x)+6(-x)^2+4(-x)^3+(-x)^4$

3. Expansion of (1+x)ⁿ and (1−x)ⁿ

Odd powers change signs, allowing coefficient sums and cancellation.

Show: Write these lines. Say: 'Keep minus x together when raising powers.'
Ask: What simplified expansion follows?
Say: Odd powers change sign: $1-4x+6x^2-4x^3+x^4$ .
Used in: Coefficient sums count coin-toss outcomes to compare chances.

If your child says…“A difference always ends with a minus.”

Ask: What sign comes from multiplying an even number of negative factors?
What’s true: The last written sign is plus for even powers and minus for odd powers.

Use the expansion

$(2x+1)^3=(2x)^3+3(2x)^2(1)+3(2x)(1)^2+1^3$

$=8x^3+3(4x^2)+6x+1$

4. Applications of the binomial theorem

Raise each complete quantity.

Show: Write these lines. Say: 'The complete quantities are twice x and one.'
Ask: What simplified cube follows?
Say: Cube the number too: $8x^3+12x^2+6x+1$ .
Used in: Volume formulas with scaled lengths help calculate storage space.

If your child says…“The power changes x, but leaves its number alone.”

Ask: How many factors of 2 does cubing twice x require?
What’s true: A power repeats the whole quantity, including its numerical factor.

$11^3=(10+1)^3$

$=10^3+3\cdot10^2+3\cdot10+1$

$=1000+300+30+1$

$=1330+1$

5. Evaluation using the binomial theorem

Split a number to make its power easier to calculate.

Show: Write these lines. Say: 'Ten has easy powers, and the adjustment is small.'
Ask: What exact total goes on the next line?
Say: Keeping every term gives the exact answer: 1331. A round number and a small adjustment make the powers easier.
Used in: Packing calculations use powers to count items arranged in equal rows and layers.

If your child says…“Every way of splitting the number makes the calculation equally easy.”

Ask: What makes the powers of ten in these lines easy?
What’s true: Valid splits give equal answers, but their powers can require different amounts of work.

Compare, divide and estimate

$(1.1)^{10}=(1+0.1)^{10}$

$=1+10(0.1)+R$

$=2+R$

6. Comparison using the binomial theorem

Positive omitted terms make a partial sum a lower bound.

Show: Write these lines for ten successive 10% increases. Call R the sum of all remaining terms.
Ask: Every term in R is positive, so how does the full power compare with 2?
Say: The remaining terms increase the total, so the power exceeds 2. We can settle this comparison without calculating every term.
Used in: Growth comparisons reveal whether repeated price increases pass a limit without calculating the final price.

If your child says…“The terms we leave out might pull the total down.”

Ask: Which factor could make any remaining term negative?
What’s true: Here every coefficient and factor is positive, so every remaining term is positive.

$3^n-2n=(1+2)^n-2n$

$=1+\sum_{r=2}^{n}\binom nr2^r$

$=1+4\sum_{r=2}^{n}\binom nr2^{r-2}$

7. Divisibility using the binomial theorem

Separate a multiple of the divisor from the remainder.

Show: Write these lines for whole-number n at least 2. Say: 'The summation sign adds the indicated terms.'
Ask: What remainder will this expression leave when divided by 4?
Say: The remainder is 1 because everything after it is an integer multiple of 4.
Used in: Remainders tell packers how many items remain after filling equal-sized packets.

If your child says…“The first terms are where the divisibility comes from.”

Ask: After cancellation, which part contains the factor 4?
What’s true: Subtracting cancels the linear term; the later terms contain the factor 4.

$(0.99)^3=(1-0.01)^3$

$=1-3(0.01)+3(0.01)^2-(0.01)^3$

$=1-0.03+0.0003-0.000001$

$\approx0.97+0.0003$

8. Approximation using the binomial theorem

Keep early terms when the omitted total is small enough.

Show: Write these lines for three successive 1% reductions. Say: 'The approximation sign means nearly equal.'
Ask: What three-term approximation goes on the next line?
Say: Keeping the signs gives approximately 0.9703. Omitting the negative final term makes this one millionth above the exact value.
Used in: Approximations help estimate a price after successive discounts.

If your child says…“I can always drop later terms because they are small.”

Ask: How much does dropping this final term change the answer?
What’s true: Check the omitted total against the accuracy needed before dropping it.

A real problem

Pascal's triangle helped solve a fair prize-sharing problem.

An interrupted dice game raised a question: how should its prize money be divided fairly?

Pascal and Fermat exchanged solutions in 1654.

Pascal's solution used his triangle.

University of St Andrews Royal Holloway Gresham College

Dice and coins beside the blank edge of an unfinished score sheet.

Concept Lab · Common misconception

The tempting mistake in Binomial Theorem

Open standalone misconception page →

Can you raise each part of a sum to a power separately?

Wrong

(x + 1)³ = x³ + 1

Why it feels right. I transfer the rule for powers of products to a sum.

Right

The expansion includes mixed terms.

Worked example

All x

xxx

Two x terms

xx1

x1x

1xx

One x term

x11

1x1

11x

All ones

111

Choose one term from each of the three factors. The mixed products occur in three different positions.

Multiply (x + 1) three times.
There are three ways to choose two x terms and three ways to choose one, giving x³ + 3x² + 3x + 1.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

What is seven choose zero?

1 There is exactly one selection containing nothing.
Expand a plus b raised to five, with a's powers decreasing. What comes third?

$10a^3b^2$ The coefficient multiplies both powers; the b part is not added separately.
Here x and y name two quantities. What term ends x minus y raised to six?

$+y^6$ Six negative factors multiply to give a positive coefficient.
What does twice x, all cubed, become?

$8x^3$ The factor 2 is multiplied three times, just like x.
Which split of 96 uses a power of ten with the smallest whole-number adjustment?

100 minus 4 Powers of 100 are easy, and subtracting 4 gives the smallest whole-number adjustment.
Does 2 overestimate or underestimate this power?

Underestimate. The omitted terms are positive and increase the total.
Take three raised to n, minus twice n. What smallest positive whole number leaves a multiple of 4 when subtracted?

1 Removing 1 leaves only an integer multiple of 4.
How far above the exact value is this three-term approximation?

0.000001 The omitted term is negative, so leaving it out raises the result.

15 more questions with answers, 1 with pictures

Teach Binomial Theorem in eight steps.

Why it matters