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24. Let S denote the multiset {o e₁,00 e2, ∞ ek}. Determine the exponential where ho 1 and, generating function for the sequence ho, h₁, h2, ..., hn, ... for n ≥ 1, " (a) hn equals the number of n-permutations of S in which each object occurs an odd number of times. (b) hn equals the number of n-permutations of S in which each object occurs at least four times. (c) hn equals the number of n-permutations of S in which e₁ occurs at least once, e2 occurs at least twice, ..., ek occurs at least k times. (d) hn equals the number of n-permutations of S in which e₁ occurs at most once, e2 occurs at most twice, ..., ek occurs at most k times.

24. Let S denote the multiset {o e₁,00 e2, ∞ ek}. Determine the exponential where ho 1 and, generating function for the sequence ho, h₁, h2, ..., hn, ... for n ≥ 1, " (a) hn equals the number of n-permutations of S in which each object occurs an odd number of times. (b) hn equals the number of n-permutations of S in which each object occurs at least four times. (c) hn equals the number of n-permutations of S in which e₁ occurs at least once, e2 occurs at least twice, ..., ek occurs at least k times. (d) hn equals the number of n-permutations of S in which e₁ occurs at most once, e2 occurs at most twice, ..., ek occurs at most k times.

Advanced Engineering Mathematics
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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24. Let S denote the multiset {∞o e₁,00 e2,...,. oek). Determine the exponential generating function for the sequence ho, h1, h2, . ..., hn, ..., where ho = 1 and, for n ≥ 1, (a) hn equals the number of n-permutations of S in which each object occurs an odd number of times. (b) hn equals the number of n-permutations of S in which each object occurs at least four times. (c) hn equals the number of n-permutations of S in which e₁ occurs at least once, e2 occurs at least twice, ..., ek occurs at least k times. (d) hn equals the number of n-permutations of S in which e₁ occurs at most once, e2 occurs at most twice, ..., ek occurs at most k times.
Transcribed Image Text:24. Let S denote the multiset {∞o e₁,00 e2,...,. oek). Determine the exponential generating function for the sequence ho, h1, h2, . ..., hn, ..., where ho = 1 and, for n ≥ 1, (a) hn equals the number of n-permutations of S in which each object occurs an odd number of times. (b) hn equals the number of n-permutations of S in which each object occurs at least four times. (c) hn equals the number of n-permutations of S in which e₁ occurs at least once, e2 occurs at least twice, ..., ek occurs at least k times. (d) hn equals the number of n-permutations of S in which e₁ occurs at most once, e2 occurs at most twice, ..., ek occurs at most k times.
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