Exercise · ZW-DERANGEMENT-010 · ★★★ Advanced
Which statement about \(D_n\) is false?
Choose one option.
Show answer
- B. \(D_n\) is even for all \(n\ge 3\).
Explanation
With \(D_4=9\), which is odd, parity fails and that counterexample is enough; the others follow from the recurrence and the limit.
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-006 In the series D_n/n!=∑_k=0^n((-1)^k)/(k!), what is the coefficient of the term k=2?
- 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?