A music festival offers has 5 stages (shows). A total of 500 people come to the festival. We are interested in the number of people who attended each of the five shows. How many possibilities are there for the attendance for each show?

The problem you presented is a classic example of distributing \(n\) identical objects (in this case, people) into \(k\) distinct groups (the stages). This is often solved using the "stars and bars" theorem, a common combinatorial technique.

Here, we have:
- \( n = 500 \) people
- \( k = 5 \) stages

We need to find the number of ways to distribute 500 people across 5 stages. The formula for this is given by:

\[
\text{Number of possibilities} = \binom{n+k-1}{k-1} = \binom{500+5-1}{5-1} = \binom{504}{4}
\]

To calculate \( \binom{504}{4} \):

\[
\binom{504}{4} = \frac{504 \times 503 \times 502 \times 501}{4 \times 3 \times 2 \times 1} 
\]

Let's calculate this value.

\[
\binom{504}{4} = \frac{504 \times 503 \times 502 \times 501}{24}
\]

First, multiply the numbers in the numerator:

\[
504 \times 503 \times 502 \times 501 = 127146374904
\]

Then divide by 24:

\[
\frac{127146374904}{24} = 5297765621
\]

Thus, the number of possibilities for the attendance for each show is \( 5,297,765,621 \).

Would you like further details on this problem or have any questions?

Here are 5 related questions:
1. How does the "stars and bars" method work in combinatorics?
2. What is the general formula for distributing \( n \) objects into \( k \) groups?
3. How would the solution change if no stage could have zero attendees?
4. What are other real-world examples of the "stars and bars" problem?
5. How does this problem relate to multinomial coefficients?

**Tip:** When dealing with combinatorial problems, identifying whether objects are distinct or identical is crucial for applying the correct formulas.

Explore how to calculate the number of ways to distribute 500 people across 5 stages at a music festival using combinatorial techniques. Learn about the stars and bars theorem and its application in solving distribution problems. Suitable for high school students studying combinatorics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation