] In a group of 90 students, each of whom has taken at least Mathematics or Computer science 

or Statistics, it was found 40 students having Mathematics, 50 students having Computer science and 

60 students having Statistics. 10 students have Mathematics as well as Computer science, 40 students 

have Mathematics as well as Statistics and 10 students have all the three subjects. Find the number of 

students who have Computer science and Statistics but not Mathematics.

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

In a group of 90 students, each of whom has taken at least Mathematics or Computer Science or Statistics, it was found that 40 students have taken Mathematics, 50 students have taken Computer Science, and 60 students have taken Statistics. 10 students have taken both Mathematics and Computer Science, 40 students have taken both Mathematics and Statistics, and 10 students have taken all three subjects. Find the number of students who have taken Computer Science and Statistics but not Mathematics.

Solve a set theory problem involving 90 students taking Mathematics, Computer Science, and Statistics. This problem uses Venn diagrams and the inclusion-exclusion principle to find how many students are enrolled in Computer Science and Statistics but not Mathematics.

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