A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

Options

A. A. 196
B. B. 206
C. C. 601
D. D. 222

Answer

To solve the problem of how many choices a student has to answer 10 out of 13 questions, with the condition that he must choose at least 4 from the first 5 questions, we can break it down into steps.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The student has to choose 10 questions from a total of 13.
- Out of the first 5 questions, he must select at least 4.

2. **Case Analysis**:
- We can break this down into two cases based on how many questions the student chooses from the first 5:
- Case 1: The student chooses 4 questions from the first 5.
- Case 2: The student chooses 5 questions from the first 5.

3. **Case 1: Choosing 4 from the first 5**:
- If the student chooses 4 questions from the first 5, he needs to choose 6 questions from the remaining 8 questions (questions 6 to 13).
- The number of ways to choose 4 questions from 5 is given by the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):
\[ \text{Ways to choose 4 from 5} = \binom{5}{4} = 5 \]
- The number of ways to choose 6 questions from 8 is:
\[ \text{Ways to choose 6 from 8} = \binom{8}{6} = \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]
- Therefore, the total number of ways for Case 1 is:
\[ 5 \times 28 = 140 \]

4. **Case 2: Choosing 5 from the first 5**:
- If the student chooses all 5 questions from the first 5, he needs to choose 5 questions from the remaining 8 questions.
- The number of ways to choose 5 questions from 5 is:
\[ \text{Ways to choose 5 from 5} = \binom{5}{5} = 1 \]
- The number of ways to choose 5 questions from 8 is:
\[ \text{Ways to choose 5 from 8} = \binom{8}{5} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]
- Therefore, the total number of ways for Case 2 is:
\[ 1 \times 56 = 56 \]

5. **Total Choices**:
- Now, we add the total choices from both cases:
\[ \text{Total Choices} = 140 + 56 = 196 \]

### Final Answer:
The total number of choices available to the student is **196**.