Find the coefficient of $x$

Scott, the coefficient is given by $\begin{pmatrix} 8 \\ n \end{pmatrix}$ as mentioned in the solution. $\begin{pmatrix} n \\ r \end{pmatrix}$ (notice I have swapped the place of $n$ ) is the combination function. Sometimes written as $^n C _r$ or $C_r^n$ . And it is defined as:

$\begin{pmatrix} n \\ r \end{pmatrix} = \frac {n!} {r!(n-r)!}$

So, we note that $\begin{pmatrix} n \\ r \end{pmatrix}$ = $\begin{pmatrix} n \\ n-r \end{pmatrix}$ . That is why the coefficients of power functions are symmetrical.

$\begin{pmatrix} n \\ r \end{pmatrix} = \dfrac {n!} {r!(n-r)!} = \dfrac {n(n-1)(n-2)...(n-r+1)} {1\times 2\times 3\times ...r}$

Therefore,

$\begin{pmatrix} 8 \\ 0 \end{pmatrix} = \begin{pmatrix} 8 \\ 8 \end{pmatrix} = 1$ -- $\begin{pmatrix} n \\ 0 \end{pmatrix}$ is defined as $1$ .

$\begin{pmatrix} 8 \\ 1 \end{pmatrix} = \begin{pmatrix} 8 \\ 7 \end{pmatrix} = \dfrac {8}{1} = 8$

$\begin{pmatrix} 8 \\ 2 \end{pmatrix} = \begin{pmatrix} 8 \\ 6 \end{pmatrix} = \dfrac {8\times 7}{1\times 2} = 28$

$\begin{pmatrix} 8 \\ 3 \end{pmatrix} = \begin{pmatrix} 8 \\ 5 \end{pmatrix} = \dfrac {8\times 7\times 6}{1\times 2\times 3} = 56$

$\begin{pmatrix} 8 \\ 4 \end{pmatrix} = \dfrac {8\times 7\times 6\times 5}{1\times 2\times 3\times 4} = 70$

If you are using Microsoft Excel spreadsheet, the function is $=COMB(n,r)$ .