Nine people sit around a round table. The number of ways of selecting four of them such that they are not from adjacent seats, is

Answer

To solve the problem of selecting 4 people from 9 seated around a round table such that no two selected people are adjacent, we can follow these steps:

### Step 1: Understand the Arrangement
Since the people are seated in a circle, we can fix one person to break the circular symmetry. This allows us to treat the problem as a linear arrangement of 8 people (the fixed person and the remaining 8).

### Step 2: Define the Selection
We need to select 4 people such that no two of them are adjacent. If we select a person, we cannot select the person immediately next to them.

### Step 3: Create Gaps
To ensure that no two selected people are adjacent, we can think of creating gaps between the selected individuals. If we select 4 people, there will be 4 gaps created by these selections. Since we have 9 total people, and we are selecting 4, we have 5 remaining people who can be placed in the gaps.

### Step 4: Distribute Remaining People
To visualize this, we can represent the selection as follows:
- Let \( x_1, x_2, x_3, x_4 \) be the selected people.
- Let \( y_1, y_2, y_3, y_4, y_5 \) be the remaining people.

We can represent the arrangement as:
- \( x_1 \) _ \( x_2 \) _ \( x_3 \) _ \( x_4 \) _
Where each underscore (_) represents a gap where we can place the remaining people.

### Step 5: Count the Gaps
To ensure that no two selected people are adjacent, we need at least one unselected person in between each selected person. This means we need to place at least one of the remaining people in each of the 4 gaps created by the selections.

### Step 6: Use Stars and Bars Method
After placing one person in each of the 4 gaps, we have used 4 of the remaining 5 people, leaving us with 1 person to place freely in any of the 5 gaps (including the ends).

Using the stars and bars method, we can find the number of ways to distribute the remaining 1 person into the 5 gaps.

### Step 7: Calculate the Combinations
The number of ways to distribute \( k \) indistinguishable items (remaining people) into \( n \) distinguishable boxes (gaps) is given by the formula:
\[ \binom{n+k-1}{k} \]
In our case, \( n = 5 \) (gaps) and \( k = 1 \) (remaining person):
\[ \binom{5+1-1}{1} = \binom{5}{1} = 5 \]

### Step 8: Final Count
Thus, the total number of ways to select 4 people from 9 seated around a round table such that no two are adjacent is \( 5 \).

### Final Answer
The number of ways of selecting four people such that they are not from adjacent seats is **5**.

---