If `\ ^(10)C_x=\ \ ^(10)C_(x+4),\ ` find the value of `xdot`

Options

A. 3
B. 7
C. 10
D. 4

Explanation

To solve the equation \( \binom{10}{x} = \binom{10}{x+4} \), we can use the property of combinations that states \( \binom{n}{r} = \binom{n}{n-r} \).

### Step-by-Step Solution:

1. **Set up the equation using the property of combinations:**
\[ \binom{10}{x} = \binom{10}{x+4} \]
According to the property, we can express \( \binom{10}{x+4} \) as:
\[ \binom{10}{x+4} = \binom{10}{10 - (x+4)} = \binom{10}{6 - x} \]
Therefore, we can rewrite the equation as:
\[ \binom{10}{x} = \binom{10}{6 - x} \]

2. **Set the indices equal to each other:**
Since the combinations are equal, we can set their indices equal:
\[ x = 6 - x \]

3. **Solve for \( x \):**
Rearranging the equation gives:
\[ x + x = 6 \]
\[ 2x = 6 \]
Dividing both sides by 2:
\[ x = 3 \]

4. **Conclusion:**
The value of \( x \) is \( 3 \).

### Final Answer:
\[ x = 3 \]