In a test paper there are total 10 questions. In how many different ways can you choose 6 questions yo answer ?
If the question number 1 is compulsory in how many ways can you choose to answer 6 questions in all ?

Options

A. 53
B. 63
C. 126
D. 210

Answer

To solve the problem of choosing 6 questions to answer from a total of 10 questions, with the condition that question number 1 is compulsory, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have 10 questions in total.
- We need to choose 6 questions to answer.
- Question number 1 is compulsory, meaning it must be included in our selection.

2. **Adjusting for the Compulsory Question**:
- Since question number 1 is compulsory, we can consider it as already chosen.
- This leaves us with 5 more questions to choose from the remaining questions.

3. **Calculating Remaining Questions**:
- After choosing question number 1, we have 9 questions left (questions 2 through 10).
- We need to choose 5 questions from these 9 remaining questions.

4. **Using Combinations**:
- The number of ways to choose 5 questions from 9 can be calculated using the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
- Here, \( n = 9 \) (the remaining questions) and \( r = 5 \) (the questions we need to choose).

5. **Calculating 9C5**:
- Plugging the values into the combination formula:
\[ 9C5 = \frac{9!}{5!(9-5)!} = \frac{9!}{5! \cdot 4!} \]
- Simplifying this:
\[ 9C5 = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \]

6. **Calculating the Numerator**:
- Calculate the numerator:
\[ 9 \times 8 = 72 \]
\[ 72 \times 7 = 504 \]
\[ 504 \times 6 = 3024 \]

7. **Calculating the Denominator**:
- Calculate the denominator:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

8. **Final Calculation**:
- Now divide the numerator by the denominator:
\[ 9C5 = \frac{3024}{24} = 126 \]

9. **Conclusion**:
- Therefore, the number of different ways to choose 6 questions to answer, with question number 1 being compulsory, is **126**.