I'm trying to prove that the coefficients $\displaystyle k_r$ for the expansion of $\displaystyle \prod_{i=1}^{n}(x-a_i)=x^n+k_{n-1}x^{n-1}+k_{n-2}x^{n-2}+\cdot\cdot\cdot+k_1x+(-1)^na_1a_2...a_n$ is

$\displaystyle k_s=(-1)^{n-s}\Bigl[\sum_{b_1<b_2<\cdot\cdot\cdot<b_{n-s}}b_1b_2\cdot\cdot\cdot b_{n-r}\Bigr]$

where $\displaystyle b_i \ \epsilon \ \{a_1,a_2,...,a_n\} $ and $\displaystyle 0<s<n$

For n = 4

$\displaystyle \prod_{i=1}^{4}(x-a_i)=$

$\displaystyle x^4-$
$\displaystyle (a_1+a_2+a_3+a_4)x^3+$
$\displaystyle (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4)x^2-$
$\displaystyle (a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4)x+$
$\displaystyle a_1a_2a_3a_4$

(Sorry for the weird way of writing, but I had to chop it up cuz I get a latex error, in the preview, saying the image is too big).

Any help would be greatly appreciated