8.1PureCore

Pascal's Triangle

Pascal’s triangle is a neat number pattern that hands you the coefficients of any binomial expansion $(a + b)^{n}$ . For small powers it is faster than any formula — just read the right row.

25 min Video by Zeeshan Zamurred The Binomial Expansion

Edexcel AS Level Maths: 8.1 Pascal’s TriangleWatch the full walkthrough before the notes below.

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What you'll be able to do

Build Pascal’s triangle row by row
Read binomial coefficients from the correct row
Use the triangle to expand (a + b)ⁿ for small n
Handle expansions with coefficients and signs

Building the triangle

Each row starts and ends with $1$ , and every other entry is the $sum of the two numbers above it$ . The rows are numbered from $0$ .

111121133114641

Row

n

gives the coefficients of

(a + b)^{n}

Reading off a row

For $(a + b)^{n}$ , use row $n$ . The powers of $a$ count down from $n$ while the powers of $b$ count up from $0$ , and the row entries are the coefficients.

1Row 3 is

1, 3, 3, 1

2Powers of

a

3, 2, 1, 0

; powers of

b

0, 1, 2, 3

Answer

a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Coefficients and signs

When the bracket contains numbers or a minus sign, substitute carefully and simplify each term. A negative second term makes the signs alternate.

1Row 3:

1, 3, 3, 1

2Terms:

2^{3} + 3 (2^{2}) (x) + 3 (2) (x^{2}) + x^{3}

Answer

8 + 12 x + 6 x^{2} + x^{3}

Tip — With a “minus” bracket like (a − b)ⁿ, the signs alternate +, −, +, − …

Formula recap

each entry = sum of the two above

Building rule.

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Row 3 in action.

a^{n} b^{0}, a^{n - 1} b^{1}, \dots, a^{0} b^{n}

Powers: a down, b up.

Common mistakes to avoid

Using the wrong row, e.g. row 4 for (a + b)³.

For (a + b)ⁿ use row n (rows are numbered from 0).

Forgetting to raise the coefficient inside the bracket to the power.

In (2 + x)³, the 2 must be cubed/squared in each term, e.g. 3(2²)x.

Key takeaways

Each entry is the sum of the two above; rows start and end with 1.
Row n gives the coefficients of (a + b)ⁿ.
Powers of a count down, powers of b count up.
Substitute carefully when the bracket has numbers or a minus sign.

Test yourself

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