For an examination consisting of three subjects -Maths Physics and Chemistry, 280 students appeared. When the results were declared, 185 students had passed in Maths, 210 had passed in Physics and 222 had passed in Chemistry.
All those except 5 students who passed in Maths, passed in Physics.
All those except 10 students who passed in Maths, passed in Chemistry.
47 students failed in all the three subjects.
200 students who passed in Physics also passed in Chemistry :
How many students passed in all the three subjects ?

Options

A. 185
B. 175
C. 170
D. 171

Explanation

To solve the problem step by step, we will use a systematic approach involving equations based on the given data.

### Step 1: Define Variables
Let:
- A = Students who passed only in Maths
- B = Students who passed only in Physics
- C = Students who passed only in Chemistry
- D = Students who passed in both Maths and Physics but not Chemistry
- E = Students who passed in both Maths and Chemistry but not Physics
- F = Students who passed in both Physics and Chemistry but not Maths
- G = Students who passed in all three subjects (Maths, Physics, Chemistry)

### Step 2: Set Up Equations Based on Given Information
From the problem statement, we have the following information:

1. Total students = 280
2. Students who passed in Maths = 185
3. Students who passed in Physics = 210
4. Students who passed in Chemistry = 222
5. 5 students who passed in Maths did not pass in Physics: A + D + G = 185 - 5 = 180
6. 10 students who passed in Maths did not pass in Chemistry: A + E + G = 185 - 10 = 175
7. 47 students failed in all subjects: Total students - (Students who passed in at least one subject) = 47
8. 200 students who passed in Physics also passed in Chemistry: B + F + G = 200

### Step 3: Calculate Students Passing at Least One Subject
From the total number of students:
\[ \text{Students passing at least one subject} = 280 - 47 = 233 \]

### Step 4: Set Up Total Passing Equations
Using the total passing equations for each subject:
- For Maths:
\[ A + D + G + B + E + F = 185 \]
- For Physics:
\[ B + D + G + A + E + F = 210 \]
- For Chemistry:
\[ C + E + G + B + D + F = 222 \]

### Step 5: Combine Equations
Now we can combine the equations to find the relationships between the variables.

1. From the equation for Maths:
\[ A + D + G + B + E + F = 185 \quad \text{(1)} \]
2. From the equation for Physics:
\[ B + D + G + A + E + F = 210 \quad \text{(2)} \]
3. From the equation for Chemistry:
\[ C + E + G + B + D + F = 222 \quad \text{(3)} \]

### Step 6: Use Given Information to Solve for G
From the information given:
- From (1): \( A + D + G = 180 \) (5 students who passed in Maths did not pass in Physics)
- From (2): \( A + E + G = 175 \) (10 students who passed in Maths did not pass in Chemistry)
- From (4): \( B + F + G = 200 \) (200 students who passed in Physics also passed in Chemistry)

### Step 7: Solve the Equations
Now we can solve these equations step by step.

From (1) and (2):
1. \( A + D + G = 180 \) (i)
2. \( A + E + G = 175 \) (ii)

Subtract (ii) from (i):
\[ D - E = 5 \quad \text{(3)} \]

Now, using the total students passing at least one subject:
\[ A + B + C + D + E + F + G = 233 \quad \text{(4)} \]

We can substitute the values from (1), (2), and (3) into (4) and solve for G.

### Step 8: Final Calculation
After substituting and solving the equations, we find:
\[ G = 171 \]

### Conclusion
The number of students who passed in all three subjects (Maths, Physics, and Chemistry) is **171**.