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?
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%.
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
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)
To find the probability of the two fiction books being all together on the left side of the shelf and the 13 nonfiction books being all together on the right side, we need to consider the total number of possible arrangements for the books.
First, let's determine the total number of possible arrangements. Since there are 15 books in total, the total number of arrangements is 15!.
Next, let's determine the number of arrangements where the two fiction books are all together on the left side and the 13 nonfiction books are all together on the right side. We can treat the two fiction books as a single entity and the 13 nonfiction books as another single entity. So, we have 2 entities to arrange.
The number of arrangements of 2 entities can be calculated as 2!.
However, within each entity, the books can be arranged in different ways. For the entity with the two fiction books, the two books can be arranged internally in 2! ways. Similarly, for the entity with the 13 nonfiction books, the books can be arranged internally in 13! ways.
Therefore, the total number of arrangements where the two fiction books are all together on the left side and the 13 nonfiction books are all together on the right side is: 2! * 2! * 13!
Finally, we can calculate the probability by dividing the number of favorable arrangements by the total number of possible arrangements.
Probability = (2! * 2! * 13!) / 15!
Simplifying this expression, we have:
Probability = (2 * 1 * 2 * 1 * 13!) / 15!
Now, we can calculate this probability using a calculator or computer software.
To find the probability of the 2 fiction books being all together on the left side and the 13 nonfiction books being all together on the right side, we need to determine the total number of possible arrangements that satisfy this condition.
First, let's consider the left side of the shelf where the fiction books are placed. Since we want these two books to be all together, we can think of them as a single unit. Hence, we can arrange these two fiction books among themselves in 2! (2 factorial) ways.
Next, let's consider the right side of the shelf where the nonfiction books are placed. Again, since we want these thirteen books to be all together, we can think of them as another single unit. Thus, we can arrange these thirteen nonfiction books among themselves in 13! (13 factorial) ways.
Now, we have arranged the left side and the right side as separate units. Therefore, the total number of arrangements is the product of the different arrangements of these two units. So the total number of arrangements satisfying the given condition is 2! * 13!.
Lastly, we need to find the total number of possible arrangements of all the fifteen books on the shelf. Since there are fifteen books in total, we can arrange them in 15! (15 factorial) ways.
To calculate the probability, we divide the number of arrangements satisfying the condition by the total number of possible arrangements: (2! * 13!) / 15!.
Now, you can simplify this expression further if you desire, and calculate the probability.