College Algebra
1st Edition
ISBN: 9781938168383
Author: Jay Abramson
Publisher: OpenStax
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Transcribed Image Text:**Package Delivery**
A delivery driver has 10 *unique* packages to deliver.
---
**i** How many ways can the delivery driver select 6 of these packages to deliver?
**ii** How many ways can the delivery driver select 4 of these packages to discard while delivering the rest? Explain how this problem relates to the first subproblem.
**iii** How many different routes may the delivery truck drive to deliver 8 of the 10 packages? A route is an ordered sequence of destinations. You may assume each package may only be delivered to its unique destination.
**iv** Before leaving for the day, the delivery trucks are loaded with packages. How many ways can 120 *unique* packages be loaded into 10 delivery trucks where some trucks may have no packages?
**v** How many ways can 120 *identical* packages be loaded into 10 delivery trucks where some trucks may have no packages?
**vi** The delivery truck driver union is concerned that the workload is unequal. How many ways can 120 *identical* packages be loaded into 10 delivery trucks where each truck must have at least 5 packages?
**vii** How many ways can 120 *unique* packages be loaded into 10 delivery trucks where each truck must have the same number of packages?
![The image shows two mathematical expressions commonly used in combinatorics:
1. **Binomial Coefficient**:
\[
\binom{n}{k} = \frac{n!}{(n-k)!k!}
\]
This expression represents the number of ways to choose \(k\) elements from a set of \(n\) elements, without considering the order of selection. Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\).
2. **Permutation**:
\[
P(n, k) = \frac{n!}{(n-k)!}
\]
This represents the number of ways to arrange \(k\) elements out of a set of \(n\) elements, taking order into account.
In these formulas:
- \(n\) is the total number of elements.
- \(k\) is the number of elements to select or arrange.
- \(n!\) denotes the factorial of \(n\), which is the product \(n \times (n-1) \times \ldots \times 1\).
- \((n-k)!\) and \(k!\) are similar factorial expressions for \(n-k\) and \(k\) respectively.](https://content.bartleby.com/qna-images/question/10732faa-f9b7-4c12-81f1-19b5d255f419/e8176671-2ce4-4dfd-af6b-873674066fe3/1vccmu_processed.png)
Transcribed Image Text:The image shows two mathematical expressions commonly used in combinatorics:
1. **Binomial Coefficient**:
\[
\binom{n}{k} = \frac{n!}{(n-k)!k!}
\]
This expression represents the number of ways to choose \(k\) elements from a set of \(n\) elements, without considering the order of selection. Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\).
2. **Permutation**:
\[
P(n, k) = \frac{n!}{(n-k)!}
\]
This represents the number of ways to arrange \(k\) elements out of a set of \(n\) elements, taking order into account.
In these formulas:
- \(n\) is the total number of elements.
- \(k\) is the number of elements to select or arrange.
- \(n!\) denotes the factorial of \(n\), which is the product \(n \times (n-1) \times \ldots \times 1\).
- \((n-k)!\) and \(k!\) are similar factorial expressions for \(n-k\) and \(k\) respectively.
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