Recall that n¡ (called subfactorial) is the number of derangements over n elements. Show that factorial and subfactorial are related by the following formulas: -Σ(0) = k=0 n! kj (Hint. Instead of using the formula for the subfactorial, you can try proving it this identity combinatorially. Note that permutations over ʼn elements can be sorted by their number of fixed elements).

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Chapter2: Second-order Linear Odes
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. Recall that n¡ (called subfactorial) is the number of derangements over n elements.
Show that factorial and subfactorial are related by the following formulas:
n
n! =E
|ki
k
k=0
(Hint. Instead of using the formula for the subfactorial, you can try proving it this
identity combinatorially. Note that permutations over n elements can be sorted by
their number of fixed elements).
Transcribed Image Text:. Recall that n¡ (called subfactorial) is the number of derangements over n elements. Show that factorial and subfactorial are related by the following formulas: n n! =E |ki k k=0 (Hint. Instead of using the formula for the subfactorial, you can try proving it this identity combinatorially. Note that permutations over n elements can be sorted by their number of fixed elements).
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