Question

The table below shows the enrollment in various classes at a certain college. 

Class	Number of Students
Biology	50
Physics	35
Calculus	40

Although no student is enrolled in all three classes, 15 are enrolled in both Biology and Physics, 10 are enrolled in both Biology and Calculus, and 12 are enrolled in both Physics and Calculus. How many students are in all three classes?

• A. 51
• B. 88
• C. 90
• D. 125
• E. 162

Hint:

Use the inclusion-exclusion principle to find the number of students in multiple sets (classes).

Answer 1

The Correct Option is A

Step 1: Use the principle of inclusion and exclusion.
Let \( x \) represent the number of students enrolled in all three classes. According to the principle of inclusion and exclusion: \[ \text{Total students in at least one class} = (\text{Biology}) + (\text{Physics}) + (\text{Calculus}) - (\text{Biology and Physics}) - (\text{Biology and Calculus}) - (\text{Physics and Calculus}) + (\text{All three classes}) \] Substitute the values: \[ 50 + 35 + 40 - 15 - 10 - 12 + x = 50 \] Step 2: Simplify and solve for \( x \).
\[ 125 - 37 + x = 50 \] \[ 88 + x = 50 \] \[ x = 50 - 88 = 51 \] Final Answer: \[ \boxed{51} \]