21. among a group of 20 students, 13 are members of the math club, 11 are members of the drama club, and 9 are members of both clubs. how many of the 20 students are not members of either club?

a. 4
b. 5
c. 11
d. 13

Question

21. among a group of 20 students, 13 are members of the math club, 11 are members of the drama club, and 9 are members of both clubs. how many of the 20 students are not members of either club?

a. 4
b. 5
c. 11
d. 13

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Venn Diagrams", "Principle of Inclusion-Exclusion" ], "current_concepts": [ "Venn Diagrams", "Principle of Inclusion-Exclusion", "Set Theory" ] } ## Identify the given sets and values Let $U$ be the universal set of students, $M$ be the set of students in the Math Club, and $D$ be the set of students in the Drama Club. $$ \begin{aligned} |U| &= 20 \ |M| &= 13 \ |D| &= 11 \ |M \cap D| &= 9 \end{aligned} $$ ## Calculate the union of the two sets Using the Principle of Inclusion-Exclusion: $$ \begin{aligned} |M \cup D| &= |M| + |D| - |M \cap D| \ |M \cup D| &= 13 + 11 - 9 \ |M \cup D| &= 15 \end{aligned} $$ ## Find the number of students not in either club Subtract the number of students in at least one club from the total number of students: $$ \begin{aligned} |U \setminus (M \cup D)| &= |U| - |M \cup D| \ &= 20 - 15 \ &= 5 \end{aligned} $$

(A) 4

(B) 5

(C) 11

(D) 13

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Set Theory" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given sets and values

Calculate the union of the two sets

Find the number of students not in either club

Answer: