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%.
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?
3 answers
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
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)
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)