Dave puts a collection of 15 books on a bookshelf in a random order. Among the books are 2 fiction and 13 nonfiction books. Whart is the probability that the 2 fiction books will be all together on the left side of the shelf and the 13 nonfiction all together on the right side of the shelf?

Answer 1

First, we need to find the total number of possible arrangements of the 15 books. This can be calculated using the formula for permutations, which is:

n! / (n-r)!

Where n is the total number of objects (15 in this case) and r is the number of objects we are selecting (also 15, since we are arranging all of the books). So, the total number of possible arrangements is:

15! / (15-15)! = 15!

Next, we need to find the number of arrangements where the 2 fiction books are together on the left side and the 13 nonfiction books are together on the right side. We can accomplish this by treating the 2 fiction books as a single unit and the 13 nonfiction books as another single unit. Then we have two units to arrange on the shelf, which can be done in 2! = 2 ways (either fiction first or nonfiction first).

Within each unit, the books can be arranged in any order. For the fiction unit, there are 2! = 2 possible arrangements, since there are 2 fiction books. For the nonfiction unit, there are 13! possible arrangements, since there are 13 nonfiction books.

So, the total number of arrangements where the 2 fiction books are together on the left side and the 13 nonfiction books are together on the right side is:

2 x 2! x 13! = 2 x 2 x 6,227,020,800 = 24,908,083,200

Finally, we can calculate the probability by dividing the number of favorable outcomes (24,908,083,200) by the total number of possible outcomes (15!), which gives:

24,908,083,200 / 15! = 24,908,083,200 / 1,307,674,368,000 = 0.019 = 1.9% (rounded to the nearest tenth)

So the probability that the 2 fiction books will be all together on the left side of the shelf and the 13 nonfiction all together on the right side of the shelf is approximately 1.9%.

Answer 2

How did you get 2 * 2! * 13! ?? Where does the extra factor of 2 come from?

As I see it, there are 2! * 13! ways to arrange them with the 2 fiction books on the left

Answer 3

You are correct, thank you for catching that mistake. We only need to multiply by 2! (not an additional 2) because there are 2! ways to arrange the 2 fiction books within the left unit. So the correct calculation should be:

2! * 13! = 2 * 6,227,020,800 = 12,454,041,600

And the final probability is:

12,454,041,600 / 15! = 12,454,041,600 / 1,307,674,368,000 = 0.00952 = 0.952% (rounded to the nearest hundredth)