We are given a group of 21 people, and we need to form a circular arrangement of 10 members, ensuring that Juan and Pedro are always included together.

Step 1: Treat Juan and Pedro as a Single Unit

Since Juan and Pedro insist on always being together, we can consider them as a single entity or a "super person." This effectively reduces the total number of people from 21 to 20.

Step 2: Selecting Members

We need to select 8 more people (since Juan and Pedro already take up 2 spots) from the remaining 19 people (since we grouped Juan and Pedro as one unit). The number of ways to choose these 8 members is:

$\binom{19}{8} = \frac{19!}{8!(11!)}$

Step 3: Arranging the Members in a Circle

For a circular arrangement, the number of ways to arrange 10 members (including Juan and Pedro as a unit) is:

$(10-1)! = 9!$

since circular permutations of $n$ elements are counted as $(n-1)!$ to avoid identical rotations being counted multiple times.

Step 4: Accounting for Juan and Pedro's Order

Since Juan and Pedro are treated as a single entity but can switch places within their unit, we must multiply by 2! to account for their internal arrangement.

Final Calculation:

$\binom{19}{8} \times 9! \times 2!$

This gives the total number of ways to form and arrange the circle while keeping Juan and Pedro together.

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Would you like a step-by-step numerical evaluation of this formula? Let me know if anything needs clarification!

Related Questions:

1. How would the solution change if Juan and Pedro could be separated?
2. What if the arrangement was a line instead of a circle?
3. How many ways could we form a circle of 15 members with the same constraints?
4. What if three specific people had to be together instead of just Juan and Pedro?
5. How does the formula change if order within the circle mattered (i.e., if the arrangement was not rotationally symmetric)?

Math Tip:

When dealing with circular permutations, always remember to subtract 1 from the number of elements before applying factorials, since rotations of the same arrangement are considered identical.