If `""^(n)C_(12)=""^(n)C_(8),` then n is equal to

Options

A. 20
B. 12
C. 6
D. 30

Answer

To solve the equation \( \binom{n}{12} = \binom{n}{8} \), we can use the property of combinations that states:

\[ \binom{n}{r} = \binom{n}{n-r} \]

### Step-by-Step Solution:

1. **Identify the Property**:
We know that \( \binom{n}{r} = \binom{n}{n-r} \). Here, we can set \( r = 8 \) and \( n - r = 12 \).

2. **Set Up the Equation**:
From the property, we have:
\[ \binom{n}{8} = \binom{n}{12} \]
This implies:
\[ n - 8 = 12 \]

3. **Solve for \( n \)**:
Rearranging the equation gives:
\[ n = 12 + 8 \]
Therefore:
\[ n = 20 \]

4. **Verify the Solution**:
We can check if our solution is correct by substituting \( n = 20 \) back into the original equation:
\[ \binom{20}{12} = \binom{20}{8} \]
Since \( \binom{n}{r} = \binom{n}{n-r} \), we have:
\[ \binom{20}{12} = \binom{20}{20-12} = \binom{20}{8} \]
Thus, our solution is verified.

### Final Answer:
The value of \( n \) is \( 20 \).