In MOCK CAT Quantitative Aptitude Section was divided into 3 groups of 5, 5 and 4 questions respectively. A candidate is required to answer 6 questions taking at least two questions from each of the first two groups and atleast one question from the third group. In how many ways can he make up his choice?

Options

A. 1200
B. 1400
C. 1284
D. 1560

Answer

To solve the problem, we need to determine the number of ways a candidate can select 6 questions from three groups, adhering to the specified conditions. Let's break it down step by step.

### Step 1: Understand the Groups and Requirements
- **Group A**: 5 questions
- **Group B**: 5 questions
- **Group C**: 4 questions
- The candidate must select a total of 6 questions with the following constraints:
- At least 2 questions from Group A
- At least 2 questions from Group B
- At least 1 question from Group C

### Step 2: Determine Possible Combinations
Given the constraints, we can have the following combinations of questions selected from each group:
1. **Case 1**: 2 from Group A, 2 from Group B, 2 from Group C
2. **Case 2**: 2 from Group A, 3 from Group B, 1 from Group C
3. **Case 3**: 3 from Group A, 2 from Group B, 1 from Group C

### Step 3: Calculate the Number of Ways for Each Case
#### Case 1: 2 from A, 2 from B, 2 from C
- Number of ways to choose 2 from Group A: \( \binom{5}{2} = 10 \)
- Number of ways to choose 2 from Group B: \( \binom{5}{2} = 10 \)
- Number of ways to choose 2 from Group C: \( \binom{4}{2} = 6 \)

Total for Case 1:
\[ 10 \times 10 \times 6 = 600 \]

#### Case 2: 2 from A, 3 from B, 1 from C
- Number of ways to choose 2 from Group A: \( \binom{5}{2} = 10 \)
- Number of ways to choose 3 from Group B: \( \binom{5}{3} = 10 \)
- Number of ways to choose 1 from Group C: \( \binom{4}{1} = 4 \)

Total for Case 2:
\[ 10 \times 10 \times 4 = 400 \]

#### Case 3: 3 from A, 2 from B, 1 from C
- Number of ways to choose 3 from Group A: \( \binom{5}{3} = 10 \)
- Number of ways to choose 2 from Group B: \( \binom{5}{2} = 10 \)
- Number of ways to choose 1 from Group C: \( \binom{4}{1} = 4 \)

Total for Case 3:
\[ 10 \times 10 \times 4 = 400 \]

### Step 4: Sum All Cases
Now, we add the total ways from all cases:
\[ 600 + 400 + 400 = 1400 \]

### Final Answer
The total number of ways the candidate can make up his choice is **1400**.

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