1. At a college library exhibition of faculty publications, 3 different mathematics books, 4 different social science books, and 3 different biology books will be displayed on a shelf. In how many ways can these ten books be arranged on the shelf if all books on the same subject are identical?
10! / (4! * 3! * 3!)
At the start of the question you say something like 3 different math books, but at the end
you say all books on the same subject are identical, so which is it?
Assuming that each subject's books are indistinguishable, we have a formula for this
number of arrangements = 10!/(3!4!3!) = .....
To determine the number of ways the ten books can be arranged on the shelf, we need to find the number of permutations of the books.
First, let's consider the mathematics books. Since all three mathematics books are identical, they can be arranged among themselves in only one way.
Next, let's consider the social science books. Since there are four different social science books, there are 4! (4 factorial) ways to arrange them among themselves.
Similarly, for the biology books, there are 3! ways to arrange them among themselves.
To find the overall number of arrangements, we can multiply the number of arrangements for each subject:
1 (mathematics) * 4! (social science) * 3! (biology)
= 1 * 4 * 3 * 2 * 1 * 3 * 2 * 1
= 1 * 4 * 6
= 24
Therefore, there are 24 ways to arrange the ten books on the shelf.
To find the number of ways the ten books can be arranged on the shelf, we need to consider the number of arrangements for each subject separately.
For the mathematics books, since they are identical, there is only one way to arrange them.
For the social science books, since they are different, we need to calculate the number of arrangements. To do this, we use the formula for permutations of distinct objects, which is n!, where n is the number of distinct objects. In this case, there are 4 social science books, so the number of arrangements is 4!.
Similarly, for the biology books, with 3 different books, the number of arrangements is 3!.
Since these three subjects are arranged independently of each other, we can multiply the number of arrangements for each subject to find the total number of arrangements.
Hence, the total number of arrangements = 1 (mathematics) * 4! (social science) * 3! (biology).
Calculating this expression, we get:
Total number of arrangements = 1 * 4! * 3!
= 1 * 24 * 6
= 144
Therefore, there are 144 different ways to arrange the ten books on the shelf if all books on the same subject are identical.