answer:Here is a simple C# class that can be used to manage a collection of books. This class uses a Dictionary to store books, where the ISBN is the key to ensure uniqueness. ```csharp using System; using System.Collections.Generic; using System.Linq; public class BookManager { private Dictionary<string, Book> _books; public BookManager() { _books = new Dictionary<string, Book>(); } public void AddBook(Book book) { if (string.IsNullOrEmpty(book.ISBN)) { throw new ArgumentException(ISBN cannot be null or empty); } if (_books.ContainsKey(book.ISBN)) { throw new ArgumentException(Book with this ISBN already exists); } _books.Add(book.ISBN, book); } public void RemoveBook(string isbn) { if (string.IsNullOrEmpty(isbn)) { throw new ArgumentException(ISBN cannot be null or empty); } if (!_books.ContainsKey(isbn)) { throw new ArgumentException(Book with this ISBN does not exist); } _books.Remove(isbn); } public void ListBooks() { foreach (var book in _books.Values) { Console.WriteLine(Title: {book.Title}, Author: {book.Author}, ISBN: {book.ISBN}); } } } ``` You can use the `BookManager` class like this: ```csharp public class Program { public static void Main() { var bookManager = new BookManager(); var book1 = new Book { Title = Book1, Author = Author1, ISBN = 123-456-789 }; bookManager.AddBook(book1); var book2 = new Book { Title = Book2, Author = Author2, ISBN = 987-654-321 }; bookManager.AddBook(book2); bookManager.ListBooks(); bookManager.RemoveBook(123-456-789); bookManager.ListBooks(); } } ``` This example creates a `BookManager` instance, adds two books, lists the books, removes one book, and lists the books again.

answer:To handle case sensitivity for ISBN numbers, you can use the `StringComparer.OrdinalIgnoreCase` when creating the dictionary, or convert the ISBN to a standard case (e.g., lower or upper case) before using it as a key in the dictionary. Here's how you can modify the `BookManager` class to handle case sensitivity: ```csharp public class BookManager { private Dictionary<string, Book> _books; public BookManager() { _books = new Dictionary<string, Book>(StringComparer.OrdinalIgnoreCase); } public void AddBook(Book book) { if (string.IsNullOrEmpty(book.ISBN)) { throw new ArgumentException(ISBN cannot be null or empty); } if (_books.ContainsKey(book.ISBN)) { throw new ArgumentException(Book with this ISBN already exists); } _books.Add(book.ISBN, book); } public void RemoveBook(string isbn) { if (string.IsNullOrEmpty(isbn)) { throw new ArgumentException(ISBN cannot be null or empty); } if (!_books.ContainsKey(isbn)) { throw new ArgumentException(Book with this ISBN does not exist); } _books.Remove(isbn); } public void ListBooks() { foreach (var book in _books.Values) { Console.WriteLine(Title: {book.Title}, Author: {book.Author}, ISBN: {book.ISBN}); } } } ``` Alternatively, you can convert the ISBN to a standard case (e.g., lower case) before using it as a key in the dictionary: ```csharp public class BookManager { private Dictionary<string, Book> _books; public BookManager() { _books = new Dictionary<string, Book>(); } public void AddBook(Book book) { if (string.IsNullOrEmpty(book.ISBN)) { throw new ArgumentException(ISBN cannot be null or empty); } var isbn = book.ISBN.ToLower(); if (_books.ContainsKey(isbn)) { throw new ArgumentException(Book with this ISBN already exists); } _books.Add(isbn, book); } public void RemoveBook(string isbn) { if (string.IsNullOrEmpty(isbn)) { throw new ArgumentException(ISBN cannot be null or empty); } var isbnLower = isbn.ToLower(); if (!_books.ContainsKey(isbnLower)) { throw new ArgumentException(Book with this ISBN does not exist); } _books.Remove(isbnLower); } public void ListBooks() { foreach (var book in _books.Values) { Console.WriteLine(Title: {book.Title}, Author: {book.Author}, ISBN: {book.ISBN}); } } } ``` Both of these approaches will ensure that the ISBN numbers are treated as case-insensitive, and duplicates will be correctly identified and prevented.

answer:Here's the adjusted code with ISBN checks modified to handle case sensitivity: ```csharp public class BookManager { private Dictionary<string, Book> _books; public BookManager() { _books = new Dictionary<string, Book>(StringComparer.OrdinalIgnoreCase); } public void AddBook(Book book) { if (string.IsNullOrWhiteSpace(book.ISBN)) { throw new ArgumentException(ISBN cannot be null, empty or whitespace); } var isbn = book.ISBN.Trim(); if (_books.ContainsKey(isbn)) { throw new ArgumentException(Book with this ISBN already exists); } _books.Add(isbn, book); } public void RemoveBook(string isbn) { if (string.IsNullOrWhiteSpace(isbn)) { throw new ArgumentException(ISBN cannot be null, empty or whitespace); } var isbnTrimmed = isbn.Trim(); if (!_books.ContainsKey(isbnTrimmed)) { throw new ArgumentException(Book with this ISBN does not exist); } _books.Remove(isbnTrimmed); } public void ListBooks() { foreach (var book in _books.Values) { Console.WriteLine(Title: {book.Title}, Author: {book.Author}, ISBN: {book.ISBN}); } } public Book GetBookByIsbn(string isbn) { if (string.IsNullOrWhiteSpace(isbn)) { throw new ArgumentException(ISBN cannot be null, empty or whitespace); } var isbnTrimmed = isbn.Trim(); if (!_books.TryGetValue(isbnTrimmed, out var book)) { throw new ArgumentException(Book with this ISBN does not exist); } return book; } } ``` However, the best approach would be using `StringComparer.OrdinalIgnoreCase` to make the dictionary lookups case insensitive as shown in the code above for `BookManager` constructor. Additionally, `Trim` is used on ISBNs to prevent any leading or trailing whitespace from affecting the lookup.

answer:binom{16}{15} = frac{16!}{15!1!} = frac{16}{1} = boxed{16}. The answer is: 16