Pascal’s triangle

Expanding (1 + x)^n, building Pascal's triangle with the addition property, its symmetry, and using a row to expand binomials such as (2 + x)^3 and (1 − 2x)^4.

The triangle

Expand $(1 + x)^n$ for $n = 0, 1, 2, \ldots$ and write down only the coefficients. Each row starts and ends with $1$.

$n = 0$	1
$n = 1$	1 1
$n = 2$	1 2 1
$n = 3$	1 3 3 1
$n = 4$	1 4 6 4 1
$n = 5$	1 5 10 10 5 1
$n = 6$	1 6 15 20 15 6 1

Each row is also symmetric: it reads the same backwards. Row $n$ has $n + 1$ entries, and we count positions from $0$.

Using a row to expand

Example 1●

Expand $(1 + x)^4$.

Row $4$: $1, 4, 6, 4, 1$. So $(1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4$.

Example 2●●

Expand $(2 + x)^3$.

$1 \cdot 2^3 + 3 \cdot 2^2 x + 3 \cdot 2x^2 + 1 \cdot x^3$

Row 3 is 1, 3, 3, 1. Powers of 2 go down while powers of x go up.

$= 8 + 12x + 6x^2 + x^3$

Example 3●●

Expand $(1 - 2x)^4$.

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

Row 4, with powers of (−2x).

$= 1 - 8x + 24x^2 - 32x^3 + 16x^4$

Flashcards

Addition property

Each entry is the sum of the two above it.

Row 5

$1, 5, 10, 10, 5, 1$

$(1 + x)^4$

$1 + 4x + 6x^2 + 4x^3 + x^4$

Practice

Find the middle entry of row $6$ of Pascal's triangle.
Find the coefficient of $x^2$ in $(1 + x)^5$.
How many entries are there in row $6$?
Row $7$ of Pascal's triangle is:
Find the coefficient of $x$ in $(2 + x)^3$.
Find the coefficient of $x^3$ in $(1 - 2x)^4$.
Find the coefficient of $x^2$ in $(1 - 2x)^4$.
$(1 + x)^3 = $
Why is each row of Pascal's triangle symmetric?