Understanding the Probability of Friends A and B Being Within Two People Apart in a Line of 12
This article explores the probability of Friends A and B being within two people apart when there are 12 individuals lined up randomly. By carefully analyzing the total arrangements and specific favorable cases, we can determine the precise likelihood of such an arrangement occurring.
Calculating the Total Arrangements
The first step involves calculating the total number of ways to arrange 12 individuals. This can be done using the factorial (denoted as !) to represent the total permutations.
Total Arrangements
The total number of arrangements of 12 individuals is given by:
Total arrangements 12!
Total arrangements 479,001,600
Evaluating Favorable Arrangements
Favorable arrangements occur when Friends A and B are not separated by more than two people. This condition can be met in three distinct ways:
Case 1: Friends A and B Are Next to Each Other (0 People in Between)
In this scenario, A and B are treated as a single unit, reducing the number of units to arrange from 12 to 11. Assuming A and B can switch places within their unit, the number of arrangements is:
11! × 2 39,916,800
Case 2: Friends A and B Have 1 Person in Between
Here, A and B form a block with one additional person in between, creating three units in total. With 10 choices for the middle person, the number of arrangements is:
10 × 10! × 2 72,576,000
Case 3: Friends A and B Have 2 People in Between
In this case, A and B are separated by two people, creating a block with five units in total. The number of ways to choose the two middle people from the remaining 10 is:
(binom{10}{2}) × 9! × 2 32,659,200
Total Favorable Arrangements
The total number of favorable arrangements is the sum of the arrangements from all three cases:
Total favorable arrangements 39,916,800 72,576,000 32,659,200 185,568,800
Probability Calculation
The probability that Friends A and B are not more than two people apart is the ratio of the total favorable arrangements to the total arrangements:
(P(text{not more than 2 in between}) frac{text{Total favorable arrangements}}{text{Total arrangements}} frac{185,568,800}{479,001,600} approx 0.3878)
This simplifies to approximately 38.78%.
Conclusion
The probability that Friends A and B are within two people apart in a line of 12 is about 0.3878 or 38.78%. This result can be valuable in understanding the probability of specific arrangements in combinatorial problems.