If ` ""^(43)C_(r-6) = ""^(43)C_(3r+1) ,` then the value of r is

Options

A. 12
B. 8
C. 6
D. 4

Answer

To solve the equation \( \binom{43}{r-6} = \binom{43}{3r+1} \), we can use the property of combinations that states \( \binom{n}{k} = \binom{n}{n-k} \).

### Step-by-Step Solution:

1. **Write the Given Equation:**
\[ \binom{43}{r-6} = \binom{43}{3r+1} \]

2. **Apply the Property of Combinations:**
According to the property, we can express \( \binom{43}{r-6} \) as:
\[ \binom{43}{r-6} = \binom{43}{43 - (r-6)} = \binom{43}{49 - r} \]
Therefore, we can rewrite the equation as:
\[ \binom{43}{49 - r} = \binom{43}{3r + 1} \]

3. **Set the Indices Equal:**
Since the combinations are equal, we can set the indices equal to each other:
\[ 49 - r = 3r + 1 \]

4. **Solve for \( r \):**
Rearranging the equation gives us:
\[ 49 - 1 = 3r + r \]
\[ 48 = 4r \]
Dividing both sides by 4:
\[ r = 12 \]

5. **Conclusion:**
The value of \( r \) is:
\[ \boxed{12} \]