Question

Each of 24 people in a movie shooting group is an actor, a musician or both. If 10 of the people in that group are actors and 19 of the group are musicians, how many of the group are musicians but not actors?

• A. \(10\)
• B. \(14\)
• C. \(19\)
• D. \(21\)

Hint:

For “either-or-both” problems, always apply the union formula first.

Answer 1

The Correct Option is B

Concept: Use the formula: \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \] where: \[ A=\text{Actors},\quad B=\text{Musicians} \]

Step 1: Substitute given values.
Total people: \[ n(A\cup B)=24 \] Actors: \[ n(A)=10 \] Musicians: \[ n(B)=19 \] So: \[ 24=10+19-n(A\cap B) \] \[ 24=29-n(A\cap B) \] \[ n(A\cap B)=5 \]

Step 2: Find musicians but not actors.
\[ n(B)-n(A\cap B) \] \[ =19-5 \] \[ =14 \] Thus, the number of musicians but not actors is: \[ \boxed{14} \]