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Chapter 3: Problem 20
If the letters in the word BOOLEAN are arranged at random, what is the probability that the two O's remain together in the arrangement?
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For each \(n \in \mathbf{Z}^{+}\)let \(B_{n}=\\{n+1, n+2, n+3, \ldots\\}\). (Here \(q=\mathbf{Z}^{+}\)and the index set \(I=\mathbf{Z}^{+}\).) Determine \(\bigcup_{n=1}^{8} B_{n}, \bigcap_{n=3}^{12} B_{n}, \bigcup_{n=1}^{\infty} B_{n}\), and \(\bigcap_{n=1}^{m} B_{n}\), where \(m\) is a fixed positive integer.
a) Determine the number of linear arrangements of \(m 1\) 's and \(r 0\) 's with no adjacent 1's. (State any needed condition(s) for \(m, r_{.}\)) b) If \(\ell=\\{1,2,3, \ldots, n\\}\), how many sets \(A \subset q_{\text {are }}\) such that \(|A|=k\) with \(A\) containing no consecutive integers? (State any needed condition(s) for \(n, k\).)
For sets \(A, B, C \subseteq Q\), prove or disprove (with a counterexample), the following: If \(A \subseteq B, B \nsubseteq C\), then \(A \nsubseteq C\).
For \(q u=\\{1,2,3, \ldots, 29,30\\}\), let \(B, C \subseteq q\) with \(B=\\{1,2,3,4,6,9,15\\}\) and \(C=\) \(\\{2,3,6,15,22,29\\}\). What is \(|B \cup C|\) ?
Darci rolls a die three times. What is the probability that a) her second and third rolls are both larger than her first roll? b) the result of her second roll is greater than that of her first roll and the result of her third roll is greater than the second?
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