Pascal's triangle and finite binomial expansion | Year 11

01 / LEARN

Understand the idea.

Understand where binomial coefficients come from

In ${(a + b)}^{n}$ , each term arises by choosing one item from each of $n$ brackets. To obtain $a^{n - r} b^{r}$ , choose the $r$ brackets supplying $b$ ; there are $(\begin{matrix} n \\ r \end{matrix})$ ways.

For a non-negative integer $n$ , this gives a finite expansion with $n + 1$ terms. Pascal's triangle supplies the coefficients: each interior entry adds the two above it, corresponding to whether a particular bracket contributes $a$ or $b$ .

{(a + b)}^{n} = \sum_{r = 0}^{n} (\begin{matrix} n \\ r \end{matrix}) a^{n - r} b^{r}

Track powers, signs and the requested term

Across the expansion, the exponent on the first term falls from $n$ to $0$ while the exponent on the second rises from $0$ to $n$ . Raise each entire term to its power, including its numerical multiplier.

For a negative second term, odd powers contribute a minus sign and even powers a plus sign. To find just one coefficient, select the term with the requested variable power and evaluate its numerical factors instead of expanding everything.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Expand with Pascal coefficients

Expand ${(x + 2)}^{4}$ .

Worked solution

1
Select the coefficient row.
$1, 4, 6, 4, 1$
The fourth power uses row 4 of Pascal's triangle, counting its top entry as row 0.
2
Insert descending and ascending powers.
$\begin{aligned} {(x + 2)}^{4} = & x^{4} + 4 x^{3} (2) + 6 x^{2} (2^{2}) \\ + 4 x (2^{3}) + 2^{4} \end{aligned}$
Each product uses four bracket choices in total, so the two exponents sum to 4.
3
Evaluate the numerical factors.
$\begin{aligned} {(x + 2)}^{4} = & x^{4} + 8 x^{3} + 24 x^{2} \\ + 32 x + 16 \end{aligned}$
The powers of 2 contribute to every coefficient as well as the constant term.

Answer

$x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$ .

EXAMPLE 02

Find one coefficient

Find the coefficient of $x^{2}$ in ${(2 x - 1)}^{5}$ .

Worked solution

1
Choose the factors needed for the requested power.
${(2 x)}^{2} {(- 1)}^{3}$
An $x^{2}$ term must use $2 x$ from exactly two brackets and $- 1$ from the remaining three.
2
Count which two brackets supply the variable.
$(\begin{matrix} 5 \\ 2 \end{matrix}) = 10$
Choosing these two positions determines all five factor choices.
3
Calculate the complete term and identify its coefficient.
$\begin{aligned} (\begin{matrix} 5 \\ 2 \end{matrix}) {(2 x)}^{2} {(- 1)}^{3} & = 10 \times 4 x^{2} \times (- 1) \\ = - 40 x^{2} \end{aligned}$
The odd power of $- 1$ is negative, so the coefficient of $x^{2}$ is $- 40$ .

Answer

−40.

03 / YOUR TURN

Try it yourself.

Check your understanding

What is the middle coefficient in ${(a + b)}^{4}$ ?

Show answer & reasoning

The five terms of ${(a + b)}^{4}$ have their middle at $a^{2} b^{2}$ . Choosing the two brackets supplying $b$ gives $(\begin{matrix} 4 \\ 2 \end{matrix}) = 6$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Write the coefficients of ${(a + b)}^{5}$ .
2 marks
Expand ${(x + 1)}^{3}$ .
2 marks
Expand ${(x - 2)}^{3}$ .
2 marks
Find the coefficient of $x^{3}$ in ${(x + 2)}^{5}$ .
2 marks
Find the constant term in ${(3 x - 2)}^{4}$ .
2 marks
Show that the sum of coefficients of ${(2 x + 1)}^{4}$ is 81.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write the coefficients of ${(a + b)}^{5}$ .
2 marks
Each interior Pascal entry is the sum of the two entries above it.
1, 5, 10, 10, 5, 1.
Expand ${(x + 1)}^{3}$ .
2 marks
Use the Pascal row 1, 3, 3, 1.
$x^{3} + 3 x^{2} + 3 x + 1$ .
Expand ${(x - 2)}^{3}$ .
2 marks
$x^{3} + 3 x^{2} (- 2) + 3 x {(- 2)}^{2} + {(- 2)}^{3}$ .
$x^{3} - 6 x^{2} + 12 x - 8$ .
Find the coefficient of $x^{3}$ in ${(x + 2)}^{5}$ .
2 marks
Choose two brackets contributing 2: $(\begin{matrix} 5 \\ 2 \end{matrix}) 2^{2} = 40$ .
40.
Find the constant term in ${(3 x - 2)}^{4}$ .
2 marks
The only constant term takes $- 2$ from all four brackets: ${(- 2)}^{4} = 16$ .
16.
Show that the sum of coefficients of ${(2 x + 1)}^{4}$ is 81.
3 marks
Setting $x$ to 1 adds the coefficients because every power of 1 is 1.
Substitute $x = 1$ : $3^{4} = 81$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.

Official curriculum source ↗

In this lesson

Understand the idea.

Understand where binomial coefficients come from

Track powers, signs and the requested term

See the method in action.

Expand with Pascal coefficients

Select the coefficient row.

Insert descending and ascending powers.

Evaluate the numerical factors.

Find one coefficient

Choose the factors needed for the requested power.

Count which two brackets supply the variable.

Calculate the complete term and identify its coefficient.

Try it yourself.

Check your understanding

Keep practising