14. Let d, be the number of derangements of n objects. For example, using the techniques of this section, we find (G) (()--)"-)) ds = 3! 1! + 2! We can use the formula for () to write this all in terms of faaorials. After simplifying, for da we would get 1. d3 = 3! 1 2. Generalize this to find a nicer formula for d. Bonus: For large n, approximately what fraction of all permutations are derangements? Use your knowledge of Taylor series from calculus.
14. Let d, be the number of derangements of n objects. For example, using the techniques of this section, we find (G) (()--)"-)) ds = 3! 1! + 2! We can use the formula for () to write this all in terms of faaorials. After simplifying, for da we would get 1. d3 = 3! 1 2. Generalize this to find a nicer formula for d. Bonus: For large n, approximately what fraction of all permutations are derangements? Use your knowledge of Taylor series from calculus.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:14. Let d, be the number of derangements of n objects. For example, using the
techniques of this section, we find
(G)
(()--)"-))
ds = 3!
1! +
2!
We can use the formula for () to write this all in terms of faaorials. After
simplifying, for da we would get
1.
d3 = 3! 1
2.
Generalize this to find a nicer formula for d. Bonus: For large n, approximately
what fraction of all permutations are derangements? Use your knowledge of
Taylor series from calculus.
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