Probability of Books Being Arranged in Specific Groups
In this article, we will explore how to calculate the probability that a collection of 11 books—5 engineering books, 4 math books, and 2 physics books—arranged on a shelf are all grouped together by their subjects. This question is a classic example of a problem involving combinatorics and permutations, which falls under the broader category of probability theory.Step-by-Step Solution
Let's break down the problem into several steps to understand how we can approach this calculation systematically.Step 1: Treating Each Category as a Single Unit
Since we want the books of each subject (engineering, math, and physics) to be together, we can treat each category as a single unit. This means we have 3 units to arrange on the shelf. 1 unit for all engineering books (5 books) 1 unit for all math books (4 books) 1 unit for all physics books (2 books)Step 2: Arranging the Units
The number of ways to arrange these 3 units is given by the factorial of the number of units, which is 3!. This can be calculated as:3! 6
Step 3: Arranging the Books Within Each Unit
Next, we need to determine the number of ways to arrange the books within each of these units. The 5 engineering books can be arranged among themselves in 5! ways, which equals 120. The 4 math books can be arranged among themselves in 4! ways, which equals 24. The 2 physics books can be arranged among themselves in 2! ways, which equals 2.Step 4: Total Arrangements of Books
To find the total number of arrangements where the books of each subject are together, we multiply the number of ways to arrange the units by the number of ways to arrange the books within each unit. Therefore, the total number of arrangements can be calculated as:Total arrangements 3! times; 5! times; 4! times; 2! 6 times; 120 times; 24 times; 2
Calculating step-by-step:
6 times; 120 720 720 times; 24 17280 17280 times; 2 34560 Thus, the total number of arrangements where the books of each subject are together is 34560.Step 5: Total Arrangements of All Books
Now, we need to calculate the total number of arrangements of the 11 books without any restrictions. This can be found by determining 11!, which is the factorial of 11.11! 39916800
Step 6: Calculating the Probability
The probability that all books of each subject are together is the ratio of the favorable arrangements to the total arrangements. Therefore, the probability P can be calculated as:P (3! times; 5! times; 4! times; 2!) / 11! 34560 / 39916800
Calculating the probability:
P 34560 / 39916800 ≈ 0.000865
Thus, the probability that all books of each subject are together is approximately:
boxed{0.000865}