Question

Question #2d003

Answer 1

`color(green)(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
`color(white)("XXX")n^4-6n^3+11n^2-6n -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
`color(white)("XXX")f(x)=x^4-6x^3+11x^2-6x-840`
`color(white)("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.