×

# How do you find the binomial expansion of (x + y)^7?

Aug 31, 2015

Use the Binomial Theorem and Pascal's triangle to find:

${\left(x + y\right)}^{7}$

$= {x}^{7} + 7 {x}^{6} y + 21 {x}^{5} {y}^{2} + 35 {x}^{4} {y}^{3} + 35 {x}^{3} {y}^{4} + 21 {x}^{2} {y}^{5} + 7 x {y}^{6} + {y}^{7}$

#### Explanation:

The Binomial Theorem tells us:

${\left(x + y\right)}^{N} = {\sum}_{n = 0}^{N} \left(\begin{matrix}N \\ n\end{matrix}\right) {x}^{N - n} {y}^{n}$

where ((N),(n)) = (N!)/(n! (N-n)!)

So in our case:

${\left(x + y\right)}^{7} = \left(\begin{matrix}7 \\ 0\end{matrix}\right) {x}^{7} + \left(\begin{matrix}7 \\ 1\end{matrix}\right) {x}^{6} y + \ldots + \left(\begin{matrix}7 \\ 6\end{matrix}\right) x {y}^{6} + \left(\begin{matrix}7 \\ 7\end{matrix}\right) {y}^{7}$

These $\left(\begin{matrix}7 \\ n\end{matrix}\right)$ coefficients occur as the 8th row of Pascal's triangle (or 7th if you choose to call the first row the 0th one as some people do). Anyway, I mean the row that starts $1 , 7$...

Each number is formed by adding the two numbers above to the left and right.

The last row gives us the coefficients we need:

${\left(x + y\right)}^{7}$

$= {x}^{7} + 7 {x}^{6} y + 21 {x}^{5} {y}^{2} + 35 {x}^{4} {y}^{3} + 35 {x}^{3} {y}^{4} + 21 {x}^{2} {y}^{5} + 7 x {y}^{6} + {y}^{7}$