In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbo: If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox? 81200439 (A) 5 (B) 7 (C) 10 (D) 17 (E) 20 (ebasanud n alione1 to 15: 25 LT
In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbo: If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox? 81200439 (A) 5 (B) 7 (C) 10 (D) 17 (E) 20 (ebasanud n alione1 to 15: 25 LT
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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![**Problem Statement:**
In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbox. If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox?
**Options:**
(A) 5
(B) 7
(C) 10
(D) 17
(E) 20
**Solution Explanation:**
To solve this problem, we can use the principle of inclusion-exclusion. Let's define:
- \( B \) as the set of students who brought backpacks.
- \( L \) as the set of students who brought lunchboxes.
According to the information given:
- \(|B| + |L| = 43\) (total number of students who brought either or both)
- \(|B| = 23\) (students who brought backpacks)
- \(|L| = 25\) (students who brought lunchboxes)
We need to find \(|B \cap L|\), the number of students who brought both a backpack and a lunchbox.
Using the formula for inclusion-exclusion:
\[
|B| + |L| - |B \cap L| = 43
\]
Substitute the known values:
\[
23 + 25 - |B \cap L| = 43
\]
Solve for \(|B \cap L|\):
\[
48 - |B \cap L| = 43
\]
\[
|B \cap L| = 48 - 43 = 5
\]
Therefore, the number of students who brought both a backpack and a lunchbox is \( \boxed{5} \), which corresponds to option (A).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F6df52e56-0a41-4469-8797-792f06ba5123%2Fo3zvx8i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbox. If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox?
**Options:**
(A) 5
(B) 7
(C) 10
(D) 17
(E) 20
**Solution Explanation:**
To solve this problem, we can use the principle of inclusion-exclusion. Let's define:
- \( B \) as the set of students who brought backpacks.
- \( L \) as the set of students who brought lunchboxes.
According to the information given:
- \(|B| + |L| = 43\) (total number of students who brought either or both)
- \(|B| = 23\) (students who brought backpacks)
- \(|L| = 25\) (students who brought lunchboxes)
We need to find \(|B \cap L|\), the number of students who brought both a backpack and a lunchbox.
Using the formula for inclusion-exclusion:
\[
|B| + |L| - |B \cap L| = 43
\]
Substitute the known values:
\[
23 + 25 - |B \cap L| = 43
\]
Solve for \(|B \cap L|\):
\[
48 - |B \cap L| = 43
\]
\[
|B \cap L| = 48 - 43 = 5
\]
Therefore, the number of students who brought both a backpack and a lunchbox is \( \boxed{5} \), which corresponds to option (A).
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