Ten people are sitting around a round table. Three of them are chosen at random to give a presentation. What is the probability that the three chosen people were sitting in consecutive seats?


To solve this problem, we need to calculate the probability that three people chosen at random from ten sitting around a circular table are sitting in consecutive seats.

### Step 1: Calculate the total number of ways to choose 3 people out of 10.
The total number of ways to choose 3 people out of 10 is given by the combination formula:

\[
\text{Total combinations} = \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]

### Step 2: Calculate the number of favorable outcomes (i.e., those where the three people are sitting in consecutive seats).
When people are seated around a circular table, the number of ways to select three consecutive people can be thought of as choosing a block of 3 consecutive seats.

- Consider one person as a reference point; there are 10 possible "starting points" for choosing a block of 3 consecutive people in a circle.
- Since it's a circle, choosing the 1st, 2nd, and 3rd seats is the same as choosing the 10th, 1st, and 2nd seats, and so on.

Therefore, there are exactly **10** different ways to choose 3 consecutive people out of 10 seated in a circle.



Explore the probability of selecting three people sitting in consecutive seats on a circular table. Learn how to calculate combinations and understand circular seating arrangements in probability. Suitable for advanced high school students.

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