**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.

The possible remainders when a number is divided by 5 are 0, 1, 2, 3, and 4. Now by pigeon hole…

Answered: How many integers from 1 through 82…

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

**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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