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in how many ways can 4 different novels, 3 different mathematics books, and 1 biology book be arranged on a bookshelf if: (a) the books can be arranged in any order? your answer is: 40320 (b) the mathematics books must be together and the novels must be together? your answer is : 864 (c) the novels must be together but the other books can be arranged in any order? your answer is

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(a) If the books can be arranged in any order, then there are 8 books in total, and each one can be placed in any of the 8 positions on the bookshelf. Therefore, the number of ways to arrange the books is:

8! = 40320

(b) If the mathematics books must be together and the novels must be together, then we can treat each group as a single "super-book" and arrange the three super-books in any order. Within each super-book, the books can be arranged in any order as well. Therefore, the number of ways to arrange the books is:

3! * 4! * 3! = 864

(c) If the novels must be together but the other books can be arranged in any order, then we can treat the four novels as a single "super-novel" and arrange the two super-books (the super-novel and the other three books) in any order. Within the super-novel, the novels can be arranged in any order as well. Therefore, the number of ways to arrange the books is:

2! * 4! = 48

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