It’s time to select next year’s housing! After frantically refreshing MyHomeAtPenn, you log onto
the website 1 minute too late to find that the three high rises are already almost full. Harrison
has enough room for 6 more people, Rodin has enough room for 5, and Harnwell has enough
for 4. Unfortunately, there are n ≥ 30 people remaining (yourself included) trying to get one
of the slots. Suddenly, the Combinatorics Genie magically appears in front of you, offering to
guarantee you a spot in one of the high rises if you can calculate how many ways the remaining
slots can be allocated. Time is ticking, so hurry and find the number of ways in which the 15
openings across the 3 high rises (6 for Harrison, 5 for Rodin, and 4 for Harnwell) can be filled
up from the pool of n prospective tenants!

This problem requires calculating the number of ways to allocate the 15 available spots across the three high rises (Harrison with 6 spots, Rodin with 5 spots, and Harnwell with 4 spots) from a pool of \( n \) prospective tenants, where \( n \geq 30 \).

This is a **combinatorics** problem where we are selecting 15 individuals from \( n \) prospective tenants. Since each high rise has a fixed number of spots, this is equivalent to partitioning 15 people into three groups of 6, 5, and 4.

### Step-by-Step Approach:

1. **Selecting the 6 tenants for Harrison:**
   - From \( n \) people, we choose 6 to live in Harrison.
   - The number of ways to do this is given by the combination:
   \[
   \binom{n}{6}
   \]

2. **Selecting the 5 tenants for Rodin:**
   - After selecting the 6 tenants for Harrison, we are left with \( n - 6 \) people.
   - From the remaining \( n - 6 \) people, we choose 5 to live in Rodin.
   - The number of ways to do this is given by the combination:
   \[
   \binom{n - 6}{5}
   \]

3. **Selecting the 4 tenants for Harnwell:**
   - After selecting the 6 tenants for Harrison and the 5 tenants for Rodin, we are left with \( n - 11 \) people.
   - From the remaining \( n - 11 \) people, we choose 4 to live in Harnwell.
   - The number of ways to do this is given by the combination:
   \[
   \binom{n - 11}{4}
   \]

4. **Final Calculation:**
   - The total number of ways to allocate the 15 slots is the product of these three combinations:
   \[
   \text{Total number of ways} = \binom{n}{6} \times \binom{n - 6}{5} \times \binom{n - 11}{4}
   \]

This formula gives the total number of ways to assign 15 people to the available spots in the three high rises.

Do you want more detailed explanations or have any specific questions about these steps?

### Relative Questions:
1. How do combinations differ from permutations in problems like this?
2. What are the properties of combinations that simplify this problem?
3. How would the solution change if the high rises had different numbers of available slots?
4. Can this problem be generalized for \( m \) high rises with varying room capacities?
5. How would the solution differ if certain individuals had preferences for specific high rises?

**Tip:** In combinatorics, always check whether the order of selection matters (for permutations) or not (for combinations) to determine the correct formula.

It’s time to select next year’s housing! After frantically refreshing MyHomeAtPenn, you log onto the website 1 minute too late to find that the three high rises are already almost full. Harrison has enough room for 6 more people, Rodin has enough room for 5, and Harnwell has enough for 4. Unfortunately, there are n ≥ 30 people remaining (yourself included) trying to get one of the slots. Suddenly, the Combinatorics Genie magically appears in front of you, offering to guarantee you a spot in one of the high rises if you can calculate how many ways the remaining slots can be allocated. Time is ticking, so hurry and find the number of ways in which the 15 openings across the 3 high rises (6 for Harrison, 5 for Rodin, and 4 for Harnwell) can be filled up from the pool of n prospective tenants!

Learn how to calculate the number of ways to allocate 15 open slots across Harrison, Rodin, and Harnwell high rises from a pool of n students. This combinatorics problem breaks down how to use binomial coefficients and combinations to find the solution. Ideal for undergraduate or advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation