The Probability of Three Particular Persons Sitting Together at a Round Table with 10 People

Imagine a scenario where 10 individuals are seated around a round table. What is the probability that 3 particular individuals will be seated together? This problem is a classic application of combinatorial probability, and it can be solved through a series of logical steps.

Introduction to the Problem

When dealing with seating arrangements around a round table, it's important to recognize that the number of ways to arrange n people in a circle is (n-1)!. This is because circular permutations are considered identical under rotations. In this case, we have 10 people, which means there are 9! ways to arrange them in a circle.

Step-by-Step Solution

Step 1: Treat the 3 Particular Persons as One Unit

The first step is to consider the 3 particular persons as a single unit or block. This transforms the problem from dealing with 10 individuals to arranging 8 units (1 block of 3 people and 7 other individuals).

Step 2: Arrange the Units in a Circle

The number of ways to arrange 8 units in a circle is given by (8-1)! 7!.

Step 3: Arrange the 3 Particular Persons Within Their Block

Since the 3 particular persons can be arranged among themselves in 3! ways, we need to multiply the number of circle arrangements by 3!

Step 4: Calculate Total Arrangements Where 3 Particular Persons Sit Together

The total number of favorable arrangements where the 3 particular persons sit together is given by:

7! times; 3!

Step 5: Calculate Total Arrangements of All Persons

The total number of ways to arrange all 10 individuals in a circle is 9!.

Step 6: Calculate the Probability

The probability that the 3 particular persons will sit together is the ratio of the number of favorable arrangements to the total arrangements:

Probability frac{7! times 3!}{9!}

Step 7: Simplify the Probability

We can simplify this expression:

Probability frac{7! times 3!}{9 times 8 times 7!} frac{3!}{9 times 8} frac{6}{72} frac{1}{12}

Final Result

Thus, the probability that the 3 particular persons will sit together at a round table with 10 people is:

P3 persons together frac{1}{12}

Conclusion

The problem of determining the probability of three particular individuals sitting together at a round table with 10 people is a great example of combinatorial probability. By breaking the problem down into smaller, manageable steps, we can arrive at the final result of frac{1}{12}.

This probability calculation is not only useful in understanding complex seating arrangements but is also applicable in various real-world scenarios, such as scheduling meetings or planning events.