In a class test 39 students passed in maths, 50 students passed in english, 39 students passed in reasoning. If 21 students passed in maths and english, 18 students passed in english and reasoning and 19 students passed in reasoning and maths, if 22 students passed in exactly two subjects then find total students who passed ?

Options

A. `82`
B. `84`
C. `66`
D. `74`

Answer

To solve the problem, we will use the principle of inclusion-exclusion and Venn diagrams. Let's break down the steps:

### Step 1: Define the Variables
Let:
- \( A \) = Number of students who passed in Maths = 39
- \( B \) = Number of students who passed in English = 50
- \( C \) = Number of students who passed in Reasoning = 39
- \( |A \cap B| \) = Number of students who passed in both Maths and English = 21
- \( |B \cap C| \) = Number of students who passed in both English and Reasoning = 18
- \( |C \cap A| \) = Number of students who passed in both Reasoning and Maths = 19
- \( |A \cap B \cap C| \) = Number of students who passed in all three subjects (to be determined)

### Step 2: Use the Information About Students Passing Exactly Two Subjects
We know that 22 students passed in exactly two subjects. This can be expressed as:
\[ |A \cap B| + |B \cap C| + |C \cap A| - 3|A \cap B \cap C| = 22 \]
Substituting the known values:
\[ 21 + 18 + 19 - 3|A \cap B \cap C| = 22 \]
Calculating the left-hand side:
\[ 58 - 3|A \cap B \cap C| = 22 \]
Rearranging gives:
\[ 3|A \cap B \cap C| = 58 - 22 \]
\[ 3|A \cap B \cap C| = 36 \]
\[ |A \cap B \cap C| = 12 \]

### Step 3: Calculate the Number of Students Passing in Each Subject
Now we can find the number of students passing in only one subject:
- Students passing only Maths:
\[ |A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) = 39 - (21 + 19 - 12) = 39 - 28 = 11 \]
- Students passing only English:
\[ |B| - (|A \cap B| + |B \cap C| - |A \cap B \cap C|) = 50 - (21 + 18 - 12) = 50 - 27 = 23 \]
- Students passing only Reasoning:
\[ |C| - (|C \cap A| + |B \cap C| - |A \cap B \cap C|) = 39 - (19 + 18 - 12) = 39 - 25 = 14 \]

### Step 4: Total Students Who Passed
Now we can find the total number of students who passed:
\[ \text{Total} = \text{Only Maths} + \text{Only English} + \text{Only Reasoning} + \text{Exactly two subjects} + \text{All three subjects} \]
Substituting the values:
\[ \text{Total} = 11 + 23 + 14 + 22 + 12 = 82 \]

### Final Answer
The total number of students who passed is **82**.

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