n! = (1) Dn + (1) Dn-1 + (2) Dn-2 + ··· + (₂²₁) D₁ + (2) Do.
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:A derangement of {1, 2, …, n} is a permutation ¿₁i2 · · · in of {1, 2, ···, n} in
which no integer is in its natural position:
i₁ ‡ 1, i2 ‡ 2, ···, in ‡ n.
Let Dn be the number of derangements of {1, 2, .. n} prove the following identity
9
n! = (1) Dn+ (1) Dn-1 + (2) Dn-2+...+(₂²₁) D₁ + (n) Do.
Here, we define Do to be 1.
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