How many distinct ways can 15 people be seated at three round tables with 5 people each if the tables are distinguishable?

To determine the number of distinct ways 15 people can be seated at three distinguishable round tables with 5 people each, we need to consider the following steps:

1. **Distribute 15 people into three groups of 5**:
   - We need to choose 5 people out of 15 for the first table, which can be done in \( \binom{15}{5} \) ways.
   - From the remaining 10 people, choose 5 for the second table, which can be done in \( \binom{10}{5} \) ways.
   - The last 5 people automatically go to the third table.

2. **Arrange each group of 5 people around a round table**:
   - For a round table, the number of distinct arrangements of 5 people is \((5-1)!\) because rotations are considered the same. So, for each table, there are \(4!\) ways to arrange the people.

Putting this together:

\[
\text{Total number of ways} = \binom{15}{5} \times \binom{10}{5} \times \binom{5}{5} \times (4!)^3
\]

Let's break this down step by step:

1. **Calculate the binomial coefficients**:
   \[
   \binom{15}{5} = \frac{15!}{5! \cdot (15-5)!} = \frac{15!}{5! \cdot 10!}
   \]
   \[
   \binom{10}{5} = \frac{10!}{5! \cdot (10-5)!} = \frac{10!}{5! \cdot 5!}
   \]
   \[
   \binom{5}{5} = \frac{5!}{5! \cdot (5-5)!} = \frac{5!}{5! \cdot 0!} = 1
   \]

2. **Calculate the factorials and their products**:
   - \(\binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003\)
   - \(\binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252\)
   - \(4! = 24\)

So the total number of ways is:
\[
3003 \times 252 \times 1 \times (24)^3
\]
\[
3003 \times 252 \times 1 \times 13824
\]
\[
3003 \times 252 \times 13824
\]

Let's calculate this step by step:

\[
3003 \times 252 = 756756
\]

\[
756756 \times 13824 = 10461394944
\]

Therefore, the total number of distinct ways to seat 15 people at three distinguishable round tables with 5 people each is \( 10,461,394,944 \).

Would you like more details or have any questions?

Here are 5 more questions you might find interesting:
1. How many ways can 10 people be seated at a round table?
2. What is the number of ways to distribute 10 different books among 4 people?
3. How many distinct ways can a group of 12 be divided into three equal groups?
4. How many ways can a committee of 5 be chosen from a group of 20 people?
5. How many permutations are there of the word "STATISTICS"?

**Tip:** When dealing with combinatorial problems, breaking the problem into smaller, manageable parts can simplify the process significantly.

Discover how to calculate the number of distinct ways to seat 15 people at three distinguishable round tables, with step-by-step guidance on using binomial coefficients and factorial arrangements. Learn about the total number of arrangements: 10,461,394,944.

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