A fixed point of a permutation f is an element x of the domain such that...
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A fixed point of a permutation f is an element x of the domain such that f(x) = x. A derangement is a permutation f with no fixed points; i.e., f(x) x for all x. (a)Prove that the probability that a random permutation f of n has f(k) = k equals 1/n. (b)If we treat the n events f(1) 1, f(n) #n as independent, what is the probability that fis a derangement? Conclude that we might expect approximately n!/e derangements of n. (c)Let D, be the number of derangements of n. Prove that the number of permutations of n with exactly k fixed points is Dn-k () - kl e: n! A fixed point of a permutation f is an element x of the domain such that f(x) = x. A derangement is a permutation f with no fixed points; i.e., f(x) x for all x. (a)Prove that the probability that a random permutation f of n has f(k) = k equals 1/n. (b)If we treat the n events f(1) 1, f(n) #n as independent, what is the probability that fis a derangement? Conclude that we might expect approximately n!/e derangements of n. (c)Let D, be the number of derangements of n. Prove that the number of permutations of n with exactly k fixed points is Dn-k () - kl e: n!
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Related Book For
Chemistry The Central Science
ISBN: 978-0321696724
12th edition
Authors: Theodore Brown, Eugene LeMay, Bruce Bursten, Catherine Murphy, Patrick Woodward
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