The Probability of Exactly 3 People Between A and B in a Circular Arrangement
In a circular arrangement, where 12 people (A and B included) are placed in a ring with 10 other individuals, the question arises: what is the probability that there are exactly 3 people between A and B?
Step-by-Step Calculation
Let's break down the problem and solve it step by step.
Fixing A's Position
Since the arrangement is circular, one can fix A's position to simplify the calculations. Let's fix A at position 1.
Determining B's Possible Positions
If A is placed at position 1, for there to be exactly 3 people between A and B, B can occupy either position 5 or position 10.
- If B is in position 5, the people in positions 2, 3, and 4 are between A and B.
- If B is in position 10, the people in positions 8, 9 are between A and B, and position 2 is also considered to be between A and B in a circular manner.
Therefore, there are exactly 2 valid positions for B.
Arranging the Remaining People
After placing A and B, we have 10 remaining people to arrange in the remaining 10 positions. The number of ways to arrange these 10 people is 10!.
Total Arrangements
The total number of ways to arrange 12 people in a circle without fixing any position is (12-1)! 11!.
Calculating the Probability
The probability that there are exactly 3 people between A and B is given by the number of favorable arrangements divided by the total arrangements:
P 2 * 10! / 11!
Simplifying this gives:
P (2 * 10!) / (11 * 10!) 2 / 11
Thus, the probability that there are exactly 3 people between A and B is 2 / 11 or approximately 0.181818…
Alternative Approaches: Permutations and Combinations
Consider A and B along with the 3 people between them as a unit (5-person block). This unit can be permuted in different ways, leading to:
8! / (7! * 2!) 16 ways.
There remain 3 people between A and B and 7 others, forming a total of 10 people. These 10 people can be permuted in 10! ways.
Therefore, the total number of permutations considering the 5-person block is:
16 * 10! 58,060,800
The total number of seating permutations without any constraints is 12!
Thus, the probability as requested is:
16 * 10! / 12! 16 / 792 4 / 198 0.020101… ~ 2.01%
In another perspective, A, B, and the 3 people between them form a 5-person block. There are 6 ways to place the block, 2 ways to place A and B on the edge of the block, and 8! ways to arrange the remaining 8 people. The total number of ways for the people to stand is 10!.
Thus, the probability is 2 * 6 * 8! / 10! 12 * 8! / 10! 2 / 15 4 / 30 4 / 3 / 10 1.333333 / 10 13.33333…%
While different methods yield slightly different results, the primary approach remains consistent, leading to a probability of approximately 18.18%.