Question

15. In your home, you have two bookshelves that will hold 10 books each and you have 25 books. You want to do the following: a. Choose 10 books to be placed on each shelf. How many sets of 10 can be chosen from the 25 books for the first shelf? b. Arrange the books on the shelfs. How many different arrangements can be made with the 10 chosen books on a shelf?

Answer 1

3,628,800 different arrangements that can be made with the 10 chosen books on a shelf.

Explanation:

a. To determine how many sets of 10 books can be chosen from the 25 books for the first shelf, we can use the combination formula. The combination formula is expressed as nCr, where n is the total number of items and r is the number of items chosen at a time.

In this case, we want to choose 10 books from a total of 25 books. So the combination formula becomes 25C10.

Using the formula, we can calculate this as:

25C10 = 25! / (10! * (25-10)!)

Simplifying further, we get:

25C10 = 25! / (10! * 15!)

Now, to calculate the value of 25!, we multiply all the numbers from 1 to 25. Similarly, we calculate the value of 10! and 15!.

After simplifying, we find that 25C10 is equal to 3,268,760.

Therefore, there are 3,268,760 sets of 10 books that can be chosen from the 25 books for the first shelf.

b. To calculate the number of different arrangements that can be made with the 10 chosen books on a shelf, we use the permutation formula. The permutation formula is expressed as nPr, where n is the total number of items and r is the number of items arranged at a time.

In this case, we want to arrange 10 books on the shelf. So the permutation formula becomes 10P10.

Using the formula, we can calculate this as:

10P10 = 10!

To calculate the value of 10!, we multiply all the numbers from 1 to 10.

After simplifying, we find that 10P10 is equal to 3,628,800.

Therefore, there are 3,628,800 different arrangements that can be made with the 10 chosen books on a shelf.