Pascal's triangle

Years 9 to 13

A triangle of numbers where each one is the two above it added together, and the rows give the numbers in a binomial expansion.

Pascal's triangle. Each number is the sum of the two above it. The highlighted row is row 4: 1, 4, 6, 4, 1.

For learners

Pascal's triangle is a pattern of numbers. Start with a 1 at the top. Every number below is made by adding the two numbers just above it. The edges are always 1.

Look at the row 1, 3, 3, 1. The 3 is made from 1 + 2 in the row above. Try it for the next row: 1, 4, 6, 4, 1.

The numbers in a row tell you how many ways there are to choose things. In the row for 4, the middle number 6 is the number of ways to choose 2 things from 4.

A worked example

Find the coefficient of $x^{2}$ in $(1 + 3 x)^{4}$ .

Row 4 of the triangle is 1, 4, 6, 4, 1.
The number for $x^{2}$ is the third one: 6.
Raise 3 to the power 2: $3^{2} = 9$ .
$6 \times 9 = 54$ . So the coefficient is 54.

For grown-ups

In the row numbered $n$ (the top 1 is row 0), the numbers are $(0 n), (1 n), \dots, (n n)$ . They are the coefficients when you expand $(1 + x)^{n}$ .

Why it matters in the questions

To expand $(1 + p x)^{n}$ , the coefficient of $x^{k}$ is the number from Pascal's triangle times $p^{k}$ . Row 4 is 1, 4, 6, 4, 1, so $(1 + 2 x)^{4} = 1 + 8 x + 24 x^{2} + 32 x^{3} + 16 x^{4}$ .

Common mistakes

Counting the top 1 as row 1. It is row 0, so the second row is row 1.
Forgetting to raise $p$ to the power as well, so only the triangle number is used.
Mixing up the numbers $(k n)$ with $n \times k$ .

Why it matters

Pascal's triangle links counting, probability and algebra. The same numbers appear when you count ways to choose, and in the binomial distribution.

Topics in quarter that use this

Binomial expansion coefficients
The binomial distribution