To solve this problem, we will use set theory and Venn diagrams to count the number of students who have Computer Science and Statistics but not Mathematics.

Let's define:

• $M$ as the set of students who have taken Mathematics.
• $C$ as the set of students who have taken Computer Science.
• $S$ as the set of students who have taken Statistics.

From the problem, we are given:

• Total number of students = 90
• $|M| = 40$ (students taking Mathematics)
• $|C| = 50$ (students taking Computer Science)
• $|S| = 60$ (students taking Statistics)
• $|M \cap C| = 10$ (students taking both Mathematics and Computer Science)
• $|M \cap S| = 40$ (students taking both Mathematics and Statistics)
• $|M \cap C \cap S| = 10$ (students taking all three subjects)

We need to find the number of students who have taken both Computer Science and Statistics but not Mathematics, which can be represented as $|C \cap S \cap M^c|$.

Step 1: Find the number of students who take Computer Science and Statistics

We know that $|M \cap C \cap S| = 10$, meaning 10 students have taken all three subjects. To find the total number of students who have taken both Computer Science and Statistics, we will use the formula: $|C \cap S| = |C| + |S| - |C \cup S|$ But $|C \cup S|$ is the union, and we can break it down by excluding the overlap with $M$. Instead, focus on how each total student population adds up.

Step 2: Substitution!

Refer back to logical separation problems<|gpt_interaction- boundary... Start