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 Maths but failed in both Physics and Chemistry ?

Options

A. 5
B. 10
C. 15
D. 1

Explanation

To solve the problem step by step, we will use the information provided to set up equations based on the number of students passing each subject and their relationships.

### Step 1: Understanding the Given Information
- Total students = 280
- Students passed in Maths (M) = 185
- Students passed in Physics (P) = 210
- Students passed in Chemistry (C) = 222
- Students failed in all subjects = 47
- Students who passed in Physics but not in Maths = 5
- Students who passed in Chemistry but not in Maths = 10
- Students who passed both Physics and Chemistry = 200

### Step 2: Calculate Students Who Passed in At Least One Subject
Since 47 students failed in all subjects, the number of students who passed in at least one subject is:
\[ 280 - 47 = 233 \]

### Step 3: Set Up Equations
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

From the information given, we can set up the following equations:

1. From Maths:
\[ a + d + e + g = 185 \quad \text{(1)} \]

2. From Physics:
\[ b + d + f + g = 210 \quad \text{(2)} \]

3. From Chemistry:
\[ c + e + f + g = 222 \quad \text{(3)} \]

4. From the relationship of students passing Physics:
\[ d + g = 205 \quad \text{(since 210 - 5 = 205)} \]

5. From the relationship of students passing Chemistry:
\[ e + g = 175 \quad \text{(since 185 - 10 = 175)} \]

6. Students passing Physics and Chemistry:
\[ f + g = 200 \quad \text{(4)} \]

### Step 4: Solve the Equations
Now we can solve these equations step by step.

From equation (4):
\[ f + g = 200 \quad \Rightarrow \quad f = 200 - g \quad \text{(5)} \]

Substituting (5) into equation (2):
\[ b + d + (200 - g) + g = 210 \]
This simplifies to:
\[ b + d + 200 = 210 \quad \Rightarrow \quad b + d = 10 \quad \text{(6)} \]

Now substituting (5) into equation (3):
\[ c + e + (200 - g) + g = 222 \]
This simplifies to:
\[ c + e + 200 = 222 \quad \Rightarrow \quad c + e = 22 \quad \text{(7)} \]

Now we have the following equations:
1. \( a + d + e + g = 185 \) (1)
2. \( b + d = 10 \) (6)
3. \( c + e = 22 \) (7)
4. \( d + g = 205 \) (4)
5. \( e + g = 175 \) (from earlier)

### Step 5: Substitute and Solve for Variables
From equation (6), we can express \( b \) in terms of \( d \):
\[ b = 10 - d \quad \text{(8)} \]

From equation (4), we can express \( g \) in terms of \( d \):
\[ g = 205 - d \quad \text{(9)} \]

Substituting (9) into equation (5):
\[ e + (205 - d) = 175 \quad \Rightarrow \quad e = 175 - 205 + d \quad \Rightarrow \quad e = d - 30 \quad \text{(10)} \]

Now substituting (10) into equation (1):
\[ a + d + (d - 30) + (205 - d) = 185 \]
This simplifies to:
\[ a + d - 30 + 205 = 185 \quad \Rightarrow \quad a + d + 175 = 185 \quad \Rightarrow \quad a + d = 10 \quad \text{(11)} \]

### Step 6: Find the Values
From equations (6) and (11), we can see that \( d = 0 \) and thus:
\[ a = 10 \quad \Rightarrow \quad e = -30 \quad \text{(not possible)} \]

### Final Calculation
We need to find students who passed in Maths but failed in both Physics and Chemistry:
\[ \text{Only Maths} = a = 10 \]

### Conclusion
Thus, the number of students who passed in Maths but failed in both Physics and Chemistry is:
\[ \boxed{1} \]