How many integers from 1 through 82 must you pick in order to be sure of getting at least two of them that have the same remainder when divided by 5? Your Answer: Answer
How many integers from 1 through 82 must you pick in order to be sure of getting at least two of them that have the same remainder when divided by 5? Your Answer: Answer
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Question:**
How many integers from 1 through 82 must you pick in order to be sure of getting at least two of them that have the same remainder when divided by 5?
**Your Answer:**
[Answer Box]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0758b56-2f3f-4908-9748-13b6d1596a48%2Fb3078277-39cc-4dcc-84a5-b99107398add%2Fzea1wo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
How many integers from 1 through 82 must you pick in order to be sure of getting at least two of them that have the same remainder when divided by 5?
**Your Answer:**
[Answer Box]
![**Problem Statement:**
You have 2 red plates, 3 green plates, and 4 blue plates. How many distinguishable ways can you stack the plates?
**Your Answer:**
[Answer Box]
---
**Explanation:**
To find the number of distinguishable ways to stack the plates, we can use the formula for permutations of multiset:
\[
\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}
\]
Where:
- \(n\) is the total number of items to arrange.
- \(n_1, n_2, \ldots, n_k\) are the frequencies of each distinct item.
In this case:
- The total number of plates \(n = 2 + 3 + 4 = 9\).
- Frequencies: 2 red plates (\(n_1 = 2\)), 3 green plates (\(n_2 = 3\)), 4 blue plates (\(n_3 = 4\)).
Substituting into the formula gives:
\[
\frac{9!}{2! \times 3! \times 4!}
\]
Calculate each factorial value and simplify to find the total number of distinguishable arrangements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0758b56-2f3f-4908-9748-13b6d1596a48%2Fb3078277-39cc-4dcc-84a5-b99107398add%2F8xyygh7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
You have 2 red plates, 3 green plates, and 4 blue plates. How many distinguishable ways can you stack the plates?
**Your Answer:**
[Answer Box]
---
**Explanation:**
To find the number of distinguishable ways to stack the plates, we can use the formula for permutations of multiset:
\[
\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}
\]
Where:
- \(n\) is the total number of items to arrange.
- \(n_1, n_2, \ldots, n_k\) are the frequencies of each distinct item.
In this case:
- The total number of plates \(n = 2 + 3 + 4 = 9\).
- Frequencies: 2 red plates (\(n_1 = 2\)), 3 green plates (\(n_2 = 3\)), 4 blue plates (\(n_3 = 4\)).
Substituting into the formula gives:
\[
\frac{9!}{2! \times 3! \times 4!}
\]
Calculate each factorial value and simplify to find the total number of distinguishable arrangements.
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