(a) The books can be arranged in any order:

This is a simple permutation problem. We have 10 different books in total (5 novels + 4 math books + 1 biology book). The number of ways to arrange n distinct items is n!, so the number of ways to arrange 10 books is 10! = 3,628,800.

(b) The mathematics books must be together and the novels must be together:

First, consider the 5 novels as one group and the 4 math books as another group. So, we have 3 groups in total (1 group of novels, 1 group of math books, and 1 biology book). These 3 groups can be arranged in 3! = 6 ways.

Within the novel group, the 5 novels can be arranged in 5! = 120 ways. Similarly, within the math book group, the 4 math books can be arranged in 4! = 24 ways.

So, the total number of arrangements is 6 * 120 * 24 = 172,800.

(c) The novels must be together but the other books can be arranged in any order:

First, consider the 5 novels as one group. So, we have 6 groups in total (1 group of novels, 4 math books, and 1 biology book). These 6 groups can be arranged in 6! = 720 ways.

Within the novel group, the 5 novels can be arranged in 5! = 120 ways.

So, the total number of arrangements is 720 * 120 = 86,400.

Question

(a) The books can be arranged in any order:

This is a simple permutation problem. We have 10 different books in total (5 novels + 4 math books + 1 biology book). The number of ways to arrange n distinct items is n!, so the number of ways to arrange 10 books is 10! = 3,628,800.

(b) The mathematics books must be together and the novels must be together:

First, consider the 5 novels as one group and the 4 math books as another group. So, we have 3 groups in total (1 group of novels, 1 group of math books, and 1 biology book). These 3 groups can be arranged in 3! = 6 ways.

Within the novel group, the 5 novels can be arranged in 5! = 120 ways. Similarly, within the math book group, the 4 math books can be arranged in 4! = 24 ways.

So, the total number of arrangements is 6 * 120 * 24 = 172,800.

(c) The novels must be together but the other books can be arranged in any order:

First, consider the 5 novels as one group. So, we have 6 groups in total (1 group of novels, 4 math books, and 1 biology book). These 6 groups can be arranged in 6! = 720 ways.

Within the novel group, the 5 novels can be arranged in 5! = 120 ways.

So, the total number of arrangements is 720 * 120 = 86,400.

Knowee AI · Accepted Answer