Question #2d003

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Question #2d003

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Answer 1 · 2016-12-20T12:40:38

$n = 7$ $color(green)(n=7)$

Explanation:

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

$n C 4 = \frac{n!}{(n - 4)! (4!)} = 35$ $nC4= (n!)/((n-4)!(4!))=35$

and since $4! = 24$ $4! =24$
$XXX \frac{n!}{n - 4}! = n \times (n - 1) \times (n - 2) \times (n - 3) = 35 \times 24$ $color(white)("XXX")(n!)/(n-4)! = nxx(n-1)xx(n-2)xx(n-3)=35xx24$

Giving
$XXX n^{4} - 6 n^{3} + 11 n^{2} - 6 n - 840 = 0$ $color(white)("XXX")n^4-6n^3+11n^2-6n -840=0$

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
$XXX f (x) = x^{4} - 6 x^{3} + 11 x^{2} - 6 x - 840$ $color(white)("XXX")f(x)=x^4-6x^3+11x^2-6x-840$
$XXXXX$ $color(white)("XXXXX")$ (Graph requires the dependent variable to be $x$ $x$ )
and found the only positive solution for $x$ $x$ was at $x = 7$ $x=7$ (or at least close to that).

Plugging $n = 7$ $n=7$ back into the polynomial verified that this solution was exact.
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Question #2d003

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