To solve this problem, we need to determine how many ways we can select 4 people from a group of 25 sitting around a circular table, ensuring that there are at least two people between each selected individual.

Step 1: Label the positions

We can represent the 25 people by labeling the seats 1 through 25 around the table.

Step 2: Applying the "at least two people between" condition

Let’s denote the positions of the 4 people we select as $p_1, p_2, p_3, p_4$, where $p_1 < p_2 < p_3 < p_4$. The condition that there are at least two people between each selected person means that:

• The distance between $p_1$ and $p_2$ must be at least 3 seats (i.e., $p_2 - p_1 \geq 3$),
• The distance between $p_2$ and $p_3$ must be at least 3 seats (i.e., $p_3 - p_2 \geq 3$),
• The distance between $p_3$ and $p_4$ must be at least 3 seats (i.e., $p_4 - p_3 \geq 3$),
• The distance between $p_4$ and $p_1$ (since it's a circular arrangement) must also be at least 3 seats (i.e., $25 - p_4 + p_1 \geq 3$).

Step 3: Simplify the problem using a transformation

To make this simpler, we perform a transformation to convert the problem into one of choosing 4 positions from a linear arrangement of people.

• Consider the 25 seats, and subtract 1 from each seat number to account for the "gap" before each selection. Then, let’s define new variables:
  • $x_1 = p_1 - 1$,
  • $x_2 = p_2 - 2$,
  • $x_3 = p_3 - 3$,
  • $x_4 = p_4 - 4$.

This transformation shifts each selected seat to ensure that the gap between them is preserved in a simpler, linear arrangement.

Now, the problem becomes selecting 4 positions from a set of 25 seats, but the positions must satisfy the condition that they are at least 3 apart. After the transformation, this is equivalent to selecting 4 distinct positions from a set of 25 positions, with no restrictions.

Step 4: Count the number of ways to select 4 people

There are 25 positions, and we need to select 4 of them, with no restrictions on the gaps between them once we’ve transformed the positions. The number of ways to select 4 positions from 25 is given by the combination formula:

$\binom{25}{4}$

Step 5: Calculate the combination

The combination formula is:

$\binom{25}{4} = \frac{25 \times 24 \times 23 \times 22}{4 \times 3 \times 2 \times 1} = 12,650$

Thus, the number of ways to select 4 people such that there are at least two people between each selected individual is 12,650.

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Would you like more detailed explanations of the steps or have any questions?

Here are 5 follow-up questions that might interest you:

1. How would this problem change if the people were arranged in a linear seating, rather than a circular one?
2. What if we wanted to select 5 people instead of 4 with the same condition?
3. How would the number of ways change if we only wanted 3 people with the "at least two people between" condition?
4. What if the number of people was increased to 30, and we still had the condition of at least two people between?
5. How can this type of problem be generalized for a table with any number of people?

Tip: When dealing with circular arrangements, it’s often helpful to think of the problem as a linear one by using transformations, as it simplifies the gap constraints.