(c) Give a 'combinatorial proof' for the following identity. That is, explain that the left hand side and right hand side are counting the same thing and so the identity is true. (3)= (-1)+(²7²) (n=¹)
(c) Give a 'combinatorial proof' for the following identity. That is, explain that the left hand side and right hand side are counting the same thing and so the identity is true. (3)= (-1)+(²7²) (n=¹)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Can you answer (2c)?
![**Problem 2:**
**(a)** How many subsets of \(\{1, 2, 3, 4, 5\}\) contain 1? Give the answer as a binomial coefficient.
**(b)** How many subsets of \(\{1, 2, 3, 4, 5\}\) do not contain 1? Give the answer as a binomial coefficient.
**(c)** Provide a ‘combinatorial proof’ for the following identity. Explain how the left-hand side and right-hand side are counting the same thing, demonstrating that the identity is true:
\[
\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}
\]
---
**Explanation:**
- A binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements.
- In (a), the task is to find the number of subsets containing the element '1'. This can be calculated by recognizing that each subset containing '1' corresponds to choosing the rest of the elements from \(\{2, 3, 4, 5\}\).
- In (b), the goal is to count subsets that do not contain '1', which involves choosing all elements from \(\{2, 3, 4, 5\}\).
- Part (c) is about understanding that this identity depicts:
- The left side, \(\binom{n}{k}\), counts the number of ways to choose \(k\) elements from \(n\) total elements.
- The identity breaks this into two cases:
- Case 1: Subsets where a specific element (like '1') is included in the subset, counted by \(\binom{n-1}{k-1}\).
- Case 2: Subsets where this element is not included, counted by \(\binom{n-1}{k}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04c291c3-b5f3-44f9-94d9-068de8214981%2F4754dee2-a8b8-4885-a0f1-11316639ca2f%2Fqzxkjbh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2:**
**(a)** How many subsets of \(\{1, 2, 3, 4, 5\}\) contain 1? Give the answer as a binomial coefficient.
**(b)** How many subsets of \(\{1, 2, 3, 4, 5\}\) do not contain 1? Give the answer as a binomial coefficient.
**(c)** Provide a ‘combinatorial proof’ for the following identity. Explain how the left-hand side and right-hand side are counting the same thing, demonstrating that the identity is true:
\[
\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}
\]
---
**Explanation:**
- A binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements.
- In (a), the task is to find the number of subsets containing the element '1'. This can be calculated by recognizing that each subset containing '1' corresponds to choosing the rest of the elements from \(\{2, 3, 4, 5\}\).
- In (b), the goal is to count subsets that do not contain '1', which involves choosing all elements from \(\{2, 3, 4, 5\}\).
- Part (c) is about understanding that this identity depicts:
- The left side, \(\binom{n}{k}\), counts the number of ways to choose \(k\) elements from \(n\) total elements.
- The identity breaks this into two cases:
- Case 1: Subsets where a specific element (like '1') is included in the subset, counted by \(\binom{n-1}{k-1}\).
- Case 2: Subsets where this element is not included, counted by \(\binom{n-1}{k}\).
Expert Solution
Step 1: Explanation - 1
We shall answer only question (2c) as asked.
We shall use the idea of combinatorics to show the identity (2c).
To do that, we will explain how both sides count the number of ways to
choose a subset of size k from a set of size n.
From the definition, we have that
choose a subset of size k from a set of size n. The left-hand side counts this by definition.
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