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Unformatted text preview: since n! thoroughly trounces 3n for n ≥ 7, as any reasonable set of data will show.
It can be shown that for any real number x > 0, not only does n! eventually overtake xn , but the
ratio x ! → 0 as n → ∞.2
Applications of factorials in the wild often involve counting arrangements. For example, if you have
ﬁfty songs on your mp3 player and wish arrange these songs in a playlist in which the order of the
2 This fact is far more important than you could ever possibly imagine. 9.4 The Binomial Theorem 583 songs matters, it turns out that there are 50! diﬀerent possible playlists. If you wish to select only
ten of the songs to create a playlist, then there are 50! such playlists. If, on the other hand, you just
want to select ten song ﬁles out of the ﬁfty to put on a ﬂash memory card so that now the order
no longer matters, there are 40!10! ways to achieve this.3 While some of these ideas are explored
in the Exercises, the authors encourage you to take courses such as Finite Mathematics, Di...
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