Question

At a college library exdibition of faculty publications, four mathentatics books, two soclat science books, and three biology books will be disptayed on a sheif: (Assume that none of the books are alike.)

(a).In how manu ways can the nine books be arranged on the shelf? ways
(b) in how mam ways can the nitie books be arranged on the sheil if books on the same subject matter are placed together?

Answer 1

The nine books can be arranged on a shelf in 362,880 ways with no restrictions. If books on the same subject are grouped together, they can be arranged in 1,728 ways.

Step-by-step explanation:

The student has asked two questions about arranging books on a shelf, which is a mathematical problem related to permutations and combinations.

Part (a):

In how many ways can nine books be arranged on a shelf?

Since there are nine different books and no restrictions, they can be arranged in 9! (9 factorial) ways. The factorial notation (n!) represents the product of all positive integers up to n. So the calculation is as follows:

9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 ways.

Part (b):

In how many ways can the nine books be arranged on the shelf if books on the same subject matter are placed together?

Here, we treat each subject group as a single item. We have three groups (mathematics, social science, and biology), which can be arranged in 3! (3 factorial) ways. Within each group, the mathematics books can be arranged in 4! ways, the social science books in 2! ways, and the biology books in 3! ways. So the total number of arrangements is:

3! x 4! x 2! x 3! = 6 x 24 x 2 x 6 = 1,728 ways.