Probability Calculation: Exactly 3 People Between A and B in a Line
This article explores the probability of having exactly 3 people between two individuals, A and B, when they stand in a line with 10 other people. It will cover various methods and approaches to solve the problem, including permutations and combinatorial mathematics.
Introduction to the Problem
The problem at hand is to calculate the probability that two individuals, A and B, stand in a line with exactly 3 people between them. We will examine three different methods to solve this problem and provide a detailed breakdown of each method.
Method 1: Direct Permutations
The first method involves direct permutations. We start by considering A and B, along with the 3 people between them, as a single block. This block can be permuted in several ways due to the internal and external permutations of A, B, and the 3 people.
Step 1: Permute A and B within the block: There are 2! ways to position A and B relative to each other within the block.
Step 2: Permute the block with the remaining 7 individuals: The block and the 7 individuals can be permuted in 10! ways.
Step 3: Permute the 3 people within the block: There are 3! ways to arrange the 3 people within the block.
Combining these, the total number of permutations is:
2! * 10! / (10! / (7! * 2!)) * 3! 2! * 8 / 12 * 11 4/33
Method 2: Block Permutations and Placement
Another approach is to consider A and B along with the 3 people between them as a single block. The block can be placed in several positions within the line, and A and B can be placed at the edges of the block.
Step 1: Determine the number of ways to form the block: There are 8! ways to arrange the 8 people in the block and the 3 individuals.
Step 2: Determine the number of ways to position the block: There are 8 ways to place the block within the 12 positions.
Step 3: Determine the number of ways to place A and B within the block: There are 2 ways to place A and B on the edges of the block.
Step 4: Determine the number of ways to arrange the remaining 7 individuals: There are 7! ways to arrange the remaining 7 individuals.
The total number of permutations is:
2! * 10! / (10! / (8! * 2!)) 2! * 16 4/33
Method 3: Positional Analysis
Another way to approach the problem is to consider the positions where A and B can be placed such that there are exactly 3 people between them. There are 6 possible positions for A and B.
Step 1: Determine the number of positions for A and B: There are 6 positions where A can be placed relative to B with 3 people in between.
Step 2: Determine the number of ways to arrange the remaining 8 people: There are 10! ways to arrange the remaining 8 people.
The required probability is:
6 * (10! / 12!) 6/132 4/33
Conclusion
The probability that there are exactly 3 people between A and B when standing in a line with 10 other people is 4/33. This result can be achieved by various methods, each involving permutations and combinatorial analysis.
The key concepts in this problem include permutations, combinatorics, and line arrangement. Understanding these concepts is crucial for solving similar probability problems in a systematic and efficient manner.
By applying different mathematical techniques, we can arrive at the same solution, providing confidence in the accuracy of our calculations.