To determine the least possible number of students who could have answered "yes" to both Question A and Question B, we can use the principle of Inclusion-Exclusion for sets.
Given data:
- Total number of students, [tex]\( |U| = 800 \)[/tex]
- Number of students answering "yes" to Question A, [tex]\( |A| = 720 \)[/tex]
- Number of students answering "yes" to Question B, [tex]\( |B| = 640 \)[/tex]
We want to find the least possible number of students who could have answered "yes" to both questions, denoted as [tex]\( |A \cap B| \)[/tex].
The principle of Inclusion-Exclusion states that:
[tex]\[ |A \cup B| = |A| + |B| - |A \cap B| \][/tex]
Here, [tex]\( |A \cup B| \)[/tex] represents the total number of students who answered "yes" to either Question A or B or both. Since the maximum number of students who responded to the survey is 800, we can have:
[tex]\[ |A \cup B| \leq 800 \][/tex]
Rewriting, we get:
[tex]\[ 800 \geq 720 + 640 - |A \cap B| \][/tex]
Solving for [tex]\( |A \cap B| \)[/tex]:
[tex]\[ 800 \geq 1360 - |A \cap B| \][/tex]
[tex]\[ |A \cap B| \geq 1360 - 800 \][/tex]
[tex]\[ |A \cap B| \geq 560 \][/tex]
Thus, the least possible number of students who could have answered "yes" to both questions is [tex]\( 560 \)[/tex].
The correct answer is:
C. 560
