Let \(A=\\{\mathrm{H}, \mathrm{T}\\}, B=\\{1,2,3,4,5,6\\},\) and \(C=\\{\) red, green blue \(\\}\). Find the numbers indicated. \(n(B \times C)\)

If you were hard pressed to study for an exam on counting and had only enough time to study one topic, would you choose the formula for the number of permutations or the multiplication principle? Give reasons for your choice.

Databases A freelance computer consultant keeps a database of her clients, which contains the names \(S=\\{\) Acme, Brothers, Crafts, Dion, Effigy, Floyd, Global, Hilbert \(\\}\) The following clients owe her money: \(A=\\{\) Acme, Crafts, Effigy, Global \(\\}\) The following clients have done at least \(\$ 10,000\) worth of business with her: \(B=\\{\) Acme, Brothers, Crafts, Dion \(\\} .\) The following clients have employed her in the last year: \(C=\\{\) Acme, Crafts, Dion, Effigy, Global, Hilbert \(\\}\). A subset of clients is described that the consultant could find using her database. Write the subset in terms of \(A, B,\) and \(C,\) and list the clients in that subset. [HINT: See Example \(4.]\) The clients who either do not owe her money, have done at least \(\$ 10,000\) worth of business with her, or have employed her in the last year

When this book was being written, the copy editor wanted to delete the comma in the following sentence (see Exercise 74 ): "You would like to see either a World War II movie, or one that is based on a comic book character but does not feature aliens." Explain why this would have resulted in an ambiguity.

Telephone Numbers Suppose a telephone number consists of a sequence of seven digits not starting with 0 or 1 . a. How many telephone numbers are possible? b. How many of them begin with either \(463,460,\) or \(400 ?\) c. How many telephone numbers are possible if no two adjacent digits are the same?

Use Venn diagrams to illustrate the given identity for subsets \(A, B,\) and \(C\) of \(S\). \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \quad\) Distributive law