In a survey of xavirs's school of Bhubaneswar of 200 students, It was found that 120 study math, 90 study physics, 70 study shemistry, 40 study math and physics, 30 study physics and chemistry, 50 study chemistry and math and 20 study none of these subjects then the number of students study all the subjects is :

Options

A. `20`
B. `30`
C. `10`
D. `18`

Explanation

To find the number of students studying all three subjects (Math, Physics, and Chemistry) in the given survey, we can use the principle of inclusion-exclusion. Let's denote:

- \( M \): Number of students studying Math = 120
- \( P \): Number of students studying Physics = 90
- \( C \): Number of students studying Chemistry = 70
- \( M \cap P \): Number of students studying both Math and Physics = 40
- \( P \cap C \): Number of students studying both Physics and Chemistry = 30
- \( C \cap M \): Number of students studying both Chemistry and Math = 50
- \( N \): Number of students studying none of the subjects = 20

### Step 1: Calculate the total number of students studying at least one subject
Total number of students surveyed = 200

Number of students studying at least one subject = Total students - Students studying none
\[ N(M \cup P \cup C) = 200 - 20 = 180 \]

### Step 2: Use the inclusion-exclusion principle
The formula for the number of students studying at least one of the subjects is given by:
\[ N(M \cup P \cup C) = N(M) + N(P) + N(C) - N(M \cap P) - N(P \cap C) - N(C \cap M) + N(M \cap P \cap C) \]

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 180 = 120 + 90 + 70 - 40 - 30 - 50 + N(M \cap P \cap C) \]

### Step 4: Simplify the equation
Calculating the right-hand side:
\[ 180 = 120 + 90 + 70 - 40 - 30 - 50 + N(M \cap P \cap C) \]
\[ 180 = 280 - 120 + N(M \cap P \cap C) \]
\[ 180 = 160 + N(M \cap P \cap C) \]

### Step 5: Solve for \( N(M \cap P \cap C) \)
Now, isolate \( N(M \cap P \cap C) \):
\[ N(M \cap P \cap C) = 180 - 160 \]
\[ N(M \cap P \cap C) = 20 \]

### Conclusion
The number of students studying all three subjects (Math, Physics, and Chemistry) is **20**.