Chapter 8: Q55E (page 513)
To determine the optimal schedule for talks, such that total number of attendees is maximized.
Therefore, the optimal schedule for talks \( = \) \(talks1,3,7\).
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Suppose that when is a positive integer divisible by 3, and . Find:
a).
b).
c)localid="1668607414775" .
A sequence \({a_1},{a_2},.....,{a_n}\) is unimodal if and only if there is an index \(m,1 \le m \le n,\) such that \({a_i} < {a_i} + 1\) when \(1{1 < i < m}\) and \({a_i} > {a_{i + 1}}\) when \(m \le i < n\). That is, the terms of the sequence strictly increase until the \(m\)th term and they strictly decrease after it, which implies that \({a_m}\) is the largest term. In this exercise, \({a_m}\) will always denote the largest term of the unimodal sequence \({a_1},{a_2},.....,{a_n}\).
a) Show that \({a_m}\) is the unique term of the sequence that is greater than both the term immediately preceding it and the term immediately following it.
b) Show that if \({a_i} < {a_i} + 1\) where \(1 \le i < n\), then \(i + 1 \le m \le n\).
c) Show that if \({a_i} > {a_{i + 1}}\) where \(1 \le i < n\), then \(1 \le m \le i\).
d) Develop a divide-and-conquer algorithm for locating the index \(m\). (Hint: Suppose that \(i < m < j\). Use parts (a), (b), and (c) to determine whether \(((i + j)/2) + 1 \le m \le n,\) \(1 \le m \le ((i + j)/2) - 1,\) or \(m = ((i + j)/2)\)
Find a closed form for the generating function for the sequence\(\left\{ {{a_n}} \right\}\), where,
a) \({a_n} = 5\) for all\(n = 0,1,2, \ldots \).
b) \({a_n} = {3^n}\)for all\(n = 0,1,2, \ldots \)
c) \({a_n} = 2\)for\(n = 3,4,5, \ldots \)and\({a_0} = {a_1} = {a_2} = 0\).
d) \({a_n} = 2n + 3\)for all\(n = 0,1,2, \ldots \)
e) \({a_n} = \left( {\begin{array}{*{20}{l}}8\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)
f) \({a_n} = \left( {\begin{array}{*{20}{c}}{n + 4}\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)
Use generating functions to find the number of ways to select 14 balls from a jar containing 100 red balls, 100 blue balls, and 100 green balls so that no fewer than 3 and no more than 10 blue balls are selected. Assume that the order in which the balls are drawn does not matter.
Find the sequence with each of these functions as its exponential generating function.
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