Probability of Two People Not Sitting Together at a Table: A Comprehensive Analysis

Understanding the probability of two specific individuals not sitting together at a round table is a classic problem in combinatorics and probability theory. This article delves into the methodology and underlying mathematical principles to derive the correct solution.

Methodology and Steps to Calculate the Probability

The problem at hand involves eight people seated around a circular table. We aim to calculate the probability that two specific people, let's call them Person A and Person B, do not sit next to each other.

Step 1: Calculate Total Arrangements

For n people seated around a circular table, the number of possible arrangements is given by (n-1)!. For this scenario with 8 people, the total possible arrangements are calculated as:

In this case, the total arrangements are (8-1)! 7! 5040.

Step 2: Calculate Arrangements with Person A and Person B Together

If Person A and Person B must sit together, we can treat them as a single unit or ldquo;blockrdquo;. This means we now have 7 units to arrange: the A-B block and the other 6 individuals.

The number of circular arrangements of these 7 units is:

Since there are 7 units, the number of arrangements is (7-1)! 6! 720.

However, since Person A and Person B can switch places within their block (A-B and B-A), we must multiply this by 2:

Thus, the total arrangements where A and B sit together are 720 x 2 1440.

Step 3: Calculate Arrangements with Person A and Person B Not Together

To find the number of arrangements where A and B do not sit together, we subtract the number of arrangements where they do sit together from the total arrangements:

Arrangements with A and B not together Total arrangements - Arrangements with A and B together 5040 - 1440 3600.

Step 4: Calculate the Probability

The probability that Person A and Person B do not sit together is the number of arrangements where they do not sit together divided by the total number of arrangements:

P(A and B not together) 3600 / 5040.

Simplifying this fraction:

3600 ÷ 720 / 5040 ÷ 720 5 / 7.

Final Answer: The probability that two specific people do not sit together at a circular table of 8 people is 5/7.

Another Method of Thinking

Alternatively, if we consider the two specified individuals as one entity, we have a total of 8 - 2 1 7 units to arrange. The favorable cases where A and B are not together can be calculated as follows:

Favorable cases 2! x (8 - 2)! 2 x 6! 2 x 720 1440. However, this does not directly give us the probability. Instead, we need to consider the unfavorable cases and use the total arrangements.

The probability of A and B not sitting together can be derived as:

Unfavorable cases (8-2) choose 2 x 2! x (8-3)! 6 choose 2 x 2 x 5! 15 x 2 x 120 3600.

Given that the total arrangements are 5040, the probability that A and B do not sit together is:

3600 / 5040 5 / 7.

Conclusion: Both methods validate the probability that two specific people do not sit together at a circular table of 8 people is 5/7.

Keywords: Probability, Circular Arrangement, Combinatorics, Seating Arrangement.