Final answer:
To arrange 8 different books (3 novels, 4 mathematics, and 1 biology book), there can be 40,320 arrangements if order doesn't matter; 864 arrangements if math books and novels are grouped; and 2,880 arrangements if only math books are grouped together.
Step-by-step explanation:
The student's question revolves around the different ways books can be arranged on a bookshelf under certain conditions. Here are the answers to each part of the question:
- Books in any order: To arrange 3 novels, 4 mathematics books, and 1 biology book in any order, you simply calculate the factorial of the total number of books, which is 8!, or 8 factorial. The factorial of a number n is the product of all positive integers less than or equal to n. Hence, the answer is 8! = 40,320.
- Mathematics books and novels together: If the mathematics books must be grouped together and the novels must be grouped together, first treat each group as a single item. This results in 3 items (1 group of novels, 1 group of math books, and 1 biology book) which can be arranged in 3! ways. Then, multiply this by the number of ways the novels can be arranged, which is 3!, and the mathematics books, which is 4!, to get the total arrangements: 3! × 3! × 4! = 6 × 6 × 24 = 864.
- Mathematics books together: When only the mathematics books need to stay together, we again treat them as a single item. We have 5 objects to arrange (1 group of math books + 3 novels + 1 biology book), which is 5!. Then multiply by the arrangements of the math books within their group, which is 4!. The answer is 5! × 4! = 120 × 24 = 2,880.
Not referenced here are the problems related to Venn diagrams, probabilities, physics of lifting books, and the logical arrangement of sentences, as they are separate and non-relevant to the question being answered.