Question #2d003

1 Answer
Dec 20, 2016

Answer:

#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.
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