Question 2d003

Dec 20, 2016

$\textcolor{g r e e n}{n = 7}$

Explanation:

Warning: This solution contains what I consider a "cheat" element.

nC4= (n!)/((n-4)!(4!))=35

and since 4! =24
color(white)("XXX")(n!)/(n-4)! = nxx(n-1)xx(n-2)xx(n-3)=35xx24#

Giving
$\textcolor{w h i t e}{\text{XXX}} {n}^{4} - 6 {n}^{3} + 11 {n}^{2} - 6 n - 840 = 0$

Here's where the "cheat" comes in. (Perhaps someone with more insight can explain how to properly factor this).

I used Graph (a free Windows program) to plot
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = {x}^{4} - 6 {x}^{3} + 11 {x}^{2} - 6 x - 840$
$\textcolor{w h i t e}{\text{XXXXX}}$(Graph requires the dependent variable to be $x$)
and found the only positive solution for $x$ was at $x = 7$ (or at least close to that).

Plugging $n = 7$ back into the polynomial verified that this solution was exact.