Your small library of books consists of 6 novels, 7 mathematics books, and 1 biology book. Calculate the number of ways to arrange them on a bookshelf for each of these situations.
a) The books can be arranged in any order?
b The novels must be together but the other books can be arranged in any order?
a) 14!
b) 6!*9!
a) To calculate the number of ways to arrange the books when they can be arranged in any order, we need to consider the total number of books in the library. In this case, we have 6 novels, 7 mathematics books, and 1 biology book. Therefore, the total number of books is 6 + 7 + 1 = 14.
When the books can be arranged in any order, it means that the order of the books does not matter. In this situation, we can use the concept of permutations, which is the arrangement of objects without repetition.
The formula to calculate the number of permutations is given by n!, where n represents the total number of objects to be arranged.
In this case, the total number of books is 14. Therefore, the number of ways to arrange them on the bookshelf is 14!.
b) Now, let's consider the situation where the novels must be together but the other books can be arranged in any order. In this case, the novels are treated as a single unit, and we need to consider the number of ways to arrange this unit along with the mathematics books and biology book.
The novels can be arranged among themselves in 6! ways, as there are 6 novels. Within this unit, their order matters.
Now, we need to consider the unit formed by the novels and find the number of ways to arrange this unit along with the mathematics books and the biology book. Since we treat the unit of novels as a single object, we can consider this as arranging a total of 8 objects (1 unit of novels, 7 mathematics books, and 1 biology book).
Using the formula for permutations, the number of ways to arrange the novels unit, mathematics books, and biology book is 8!.
Therefore, the number of ways to arrange the books in this situation is 6! × 8!.
Note: In both cases, the exclamation mark (!) denotes the factorial operation, which is the product of all positive integers from 1 to n.