Exercise · ZW-DERANGEMENT-006 · ★★☆ Intermediate
In the series \(D_n/n!=\sum_{k=0}^{n}\frac{(-1)^k}{k!}\), what is the coefficient of the term \(k=2\)?
Choose one option.
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- B. \(1/2\)
Explanation
The term \(k=2\) is \((-1)^2/2!\); since \((-1)^2=1\), the coefficient is \(+1/2\).
Review the concept: Derangement
A derangement is a permutation of \(n\) elements with no fixed points; the number of derangements is \(D_n\).
\[D_n = n!\sum_{k=0}^{n}\frac{(-1)^k}{k!}\]
More practice problems: derangement
- ZW-DERANGEMENT-001 Which of these permutations of \1,2,3,4\, written as σ(1)σ(2)σ(3)σ(4), is a derangement?
- ZW-DERANGEMENT-002 Of the 9 derangements of 4 elements, how many are a single 4-cycle?
- ZW-DERANGEMENT-003 What is D_3?
- ZW-DERANGEMENT-004 What is D_4?
- ZW-DERANGEMENT-005 If D_n/n! approaches 1/e, what approximate proportion of permutations of n elements has at least on…
- ZW-DERANGEMENT-007 For large n, what value does D_n/n! approach?
- ZW-DERANGEMENT-008 How many permutations of 5 elements have exactly one fixed point?
- ZW-DERANGEMENT-009 How many permutations of 4 elements fix at least one point?