what is the value of x when `""^(11) C_x ` for any natural number x ?

Options

A. a. 7
B. b. 6
C. d. 5
D. d. both (b) and (c )

Answer

To find the value of \( x \) such that \( \binom{11}{x} \) is a natural number, we can analyze the combinations formula and check for various values of \( x \).

### Step-by-Step Solution:

1. **Understanding the Combinations Formula**:
The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items, and \( r \) is the number of items to choose.

2. **Applying the Formula**:
In our case, \( n = 11 \). We need to find \( x \) such that \( \binom{11}{x} \) is a natural number. This means we need to check values of \( x \) from 0 to 11.

3. **Calculating for Different Values of \( x \)**:
- **For \( x = 0 \)**:
\[ \binom{11}{0} = 1 \quad (\text{natural number}) \]
- **For \( x = 1 \)**:
\[ \binom{11}{1} = 11 \quad (\text{natural number}) \]
- **For \( x = 2 \)**:
\[ \binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55 \quad (\text{natural number}) \]
- **For \( x = 3 \)**:
\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \quad (\text{natural number}) \]
- **For \( x = 4 \)**:
\[ \binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330 \quad (\text{natural number}) \]
- **For \( x = 5 \)**:
\[ \binom{11}{5} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462 \quad (\text{natural number}) \]
- **For \( x = 6 \)**:
\[ \binom{11}{6} = \binom{11}{5} = 462 \quad (\text{natural number}) \]
- **For \( x = 7 \)**:
\[ \binom{11}{7} = \binom{11}{4} = 330 \quad (\text{natural number}) \]
- **For \( x = 8 \)**:
\[ \binom{11}{8} = \binom{11}{3} = 165 \quad (\text{natural number}) \]
- **For \( x = 9 \)**:
\[ \binom{11}{9} = \binom{11}{2} = 55 \quad (\text{natural number}) \]
- **For \( x = 10 \)**:
\[ \binom{11}{10} = 11 \quad (\text{natural number}) \]
- **For \( x = 11 \)**:
\[ \binom{11}{11} = 1 \quad (\text{natural number}) \]

4. **Conclusion**:
Since \( \binom{11}{x} \) is a natural number for all \( x \) in the range \( 0 \leq x \leq 11 \), the values of \( x \) can be any natural number from 0 to 11.

### Final Answer:
The value of \( x \) can be any natural number from 0 to 11.