91Ó°ÊÓ

How many permutations of the 26 different letters of the alphabet contain (a) either the pattern "OUT" or the pattern "DIG"? (b) neither the pattern "MAN" nor the pattem "ANT"?

A set \(A\) has 128 subsets of even cardinality. (a) How many subsets of \(A\) have odd cardinality? (b) What is \(|A|\) ?

For \(A=\\{1,2,3,4,5,6,7\\}\), determine the number of a) subsets of \(A\). b) nonempty subsets of \(A\). c) proper subsets of \(A\). d) nonempty proper subsets of \(A\). e) subsets of \(A\) containing three elements. f) subsets of \(A\) containing 1,2 . g) subsets of \(A\) containing five elements, including \(1,2 .\) h) proper subsets of \(A\) containing 1,2 . i) subsets of \(A\) with an even number of elements. j) subsets of \(A\) with an odd number of elements. k) subsets of \(A\) with an odd number of elements, including the element \(3 .\)

Let \(A=\\{1,2,3, \ldots, 15\\}\) a) How many subsets of \(A\) contain all of the odd integers in \(A\) ? b) How many subsets of \(A\) contain exactly three odd integers? c) How many eight-element subsets of \(A\) contain exactly three odd integers? d) Write a computer program (or develop an algorithm) to generate an eight- element subset of \(A\) and have it print out how many of the eight elements are odd.

a) If a set \(A\) has 63 proper subsets, what is \(|A| ?\) b) If a set \(B\) has 64 subsets of odd cardinality, what is \(|B|\) ? c) Generalize the result of part (b).

When he is about to leave a restaurant counter, a man sees that he has one penny, oot nickel, one dime, one quarter, and one half-dollar. In how many ways can he leave some (at least one) of his coins for a tip if (a) there are no restrictions? (b) he wants to have some change left? (c) he wants to leave at least 10 cents?

Prove or disprove each of the following: a) For sets \(A, B, C \subseteq q, A \cap C=B \cap C \Rightarrow A=B\). b) For sets \(A, B, C \subseteq q, A \cup C=B \cup C \Rightarrow A=B\). c) For sets \(A, B, C \subseteq ?,[(A \cap C=B \cap C) \wedge(A \cup C=B \cup C)] \Rightarrow A=B\). d) For sets \(A, B, C \subseteq q, A \Delta C=B \Delta C \Rightarrow A=B\). &\text { b) } A \cap(B \Delta C)=(A \cap B) \Delta(A \cap C) \\ &\text { c) } A \triangle(B \cup C)=(A \triangle B) \cup(A \triangle C) \\ &\text { d) } A-(B \cup C)=(A-B) \cap(A-C) \\ &\text { e) } A \Delta(B \triangle C)=(A \triangle B) \triangle C \end{aligned} $$

a) How many arrangements of the letters in MISCELLANEOUS have no pair of consecutive identical letters? b) If an arrangement of these letters is randomly generated, what is the probability that no pair of consecutive identical letters occurs?

a) How many subsets of \(\\{1,2,3, \ldots, 11\\}\) contain at least one even integer? b) How many subsets of \(\\{1,2,3, \ldots, 12\\}\) contain at least one even integer? c) Generalize the results of parts (a) and (b).

Use membership tables to establish each of the following: a) \(\overline{A \cap B}=\bar{A} \cup \bar{B}\) b) \(A \cup A=A\) c) \(A \cup(A \cap B)=A\) d) \(\frac{A \cap \bar{B}-\bar{A} \cup \bar{B}}{(A \cap B) \cup(\bar{A} \cap C)}=(A \cap \bar{B}) \cup(\bar{A} \cap \bar{C})\)