There are three papers of 100 marks each in an examination. Then the no. of ways can a student get 150 marks such that he gets atleast `60%` in two papers

Options

A. `""^(3)C_(2)xx""^(32)C_(2)`
B. `""^(4)C_(3)xx""^(32)C_(2)`
C. `""^(4)C_(3)xx""^(36)C_(2)`
D. `""^(4)C_(3)xx""^(36)C_(3)`

Answer

To solve the problem of how many ways a student can score 150 marks in three papers of 100 marks each, with the condition that they score at least 60% (i.e., at least 60 marks) in two of the papers, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Each paper is out of 100 marks.
- The total marks obtained by the student is 150.
- The student must score at least 60 marks in two papers.

2. **Setting Up the Scores**:
- Let the scores in the three papers be \( x_1, x_2, x_3 \).
- We know that \( x_1 + x_2 + x_3 = 150 \).
- The condition requires that at least two of the scores \( x_1, x_2, x_3 \) are at least 60.

3. **Case Analysis**:
- We can have two cases based on how the scores are distributed:
- **Case 1**: Two papers score 60 marks each, and the third paper scores 30 marks.
- **Case 2**: One paper scores 60 marks, and the other two papers score more than 60 marks.

4. **Case 1 Calculation**:
- For \( x_1 = 60, x_2 = 60, x_3 = 30 \):
- The number of ways to assign the scores is given by the permutations of the scores.
- The scores can be arranged in \( \frac{3!}{2!} = 3 \) ways (since two scores are the same).

5. **Case 2 Calculation**:
- For this case, we can have scores like \( x_1 = 60, x_2 = a, x_3 = b \) where \( a + b = 90 \) and \( a, b \geq 60 \).
- The scores can be \( 61, 29 \), \( 62, 28 \), ..., \( 100, 0 \).
- This means \( a \) can take values from 60 to 90, giving us \( 31 \) options (from 60 to 90 inclusive).

6. **Combining Cases**:
- For Case 1, we have 3 ways.
- For Case 2, we need to consider how many ways we can choose two papers to score at least 60 marks.
- The number of ways to choose 2 papers out of 3 is \( \binom{3}{2} = 3 \).

7. **Final Calculation**:
- The total number of ways is:
\[ \text{Total Ways} = \text{Ways from Case 1} + \text{Ways from Case 2} \]
\[ = 3 + 3 \cdot 31 = 3 + 93 = 96 \]

### Conclusion:
The total number of ways a student can score 150 marks such that they get at least 60% in two papers is **96**.