The class has thirty students. Twenty students study physics while twelve study both physics and chemistry. If eight students study neither physics nor chemistry, how many students... The class has thirty students. Twenty students study physics while twelve study both physics and chemistry. If eight students study neither physics nor chemistry, how many students study chemistry?
Understand the Problem
The question asks to find the number of students studying chemistry given the total number of students, the number of students studying physics, the number of students studying both physics and chemistry, and the number of students studying neither.
Answer
14
Answer for screen readers
14
Steps to Solve
- Find the number of students studying either physics or chemistry or both
Since there are 30 students in total and 8 study neither subject, the number of students studying at least one subject is:
$30 - 8 = 22$
- Use the principle of inclusion-exclusion to find the number of students studying chemistry
Let P be the number of students studying physics, C be the number of students studying chemistry, and $P \cap C$ be the number of students studying both. We are given:
$|P| = 20$ $|P \cap C| = 12$ $|P \cup C| = 22$
We know that:
$|P \cup C| = |P| + |C| - |P \cap C|$
Substituting the given values:
$22 = 20 + |C| - 12$
- Solve for $|C|$
$22 = 8 + |C|$
$|C| = 22 - 8$
$|C| = 14$
Therefore, the number of students studying chemistry is 14.
14
More Information
The principle of inclusion-exclusion is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two sets
Tips
A common mistake is forgetting to account for the overlap (students studying both subjects) when using the inclusion-exclusion principle.
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