Chapter 8: Q41SE (page 567)
What is the probability that exactly one person is given back the correct hat by a hatcheck person who gives n people their hats back at random?
The probability that exactly one person is given back the correct hat is \(\frac{{n{D_{n - 1}}}}{{n!}} = \frac{{{D_{n - 1}}}}{{(n - 1)!}}\)
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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\(n = 0,1,2, \ldots \)
Express the fast multiplication algorithm in pseudocode.
Prove Theorem 4.
Find the coefficient of \({x^9}\) in the power series of each of these functions.
a) \({\left( {1 + {x^3} + {x^6} + {x^9} + \cdots } \right)^3}\)
b) \({\left( {{x^2} + {x^3} + {x^4} + {x^5} + {x^6} + \cdots } \right)^3}\)
c) \(\left( {{x^3} + {x^5} + {x^6}} \right)\left( {{x^3} + {x^4}} \right)\left( {x + {x^2} + {x^3} + {x^4} + \cdots } \right)\)
d) \(\left( {x + {x^4} + {x^7} + {x^{10}} + \cdots } \right)\left( {{x^2} + {x^4} + {x^6} + {x^8} + } \right.\)\( \cdots )\)
e) \({\left( {1 + x + {x^2}} \right)^3}\)
Find a closed form for the generating function for the sequence\(\left\{ {{a_n}} \right\}\), where
a) \({a_n} = - 1\) for all\(n = 0,1,2, \ldots \).
b) \({a_n} = {2^n}\)for\(n = 1,2,3,4, \ldots \)and\({a_0} = 0\).
c) \({a_n} = n - 1\)for\(n = 0,1,2, \ldots \).
d) \({a_n} = 1/(n + 1)!\)for\(n = 0,1,2, \ldots \)
e) \({a_n} = \left( {\begin{array}{*{20}{l}}n\\2\end{array}} \right)\)for\(n = 0,1,2, \ldots \)
f) \({a_n} = \left( {\begin{array}{*{20}{c}}{10}\\{n + 1}\end{array}} \right)\)for\(n = 0,1,2, \ldots \)
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