If `\ ^n C_(12)=\ \ ^n C_8` then `n=`

Options

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

Explanation

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

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

This means that \(\binom{n}{12} = \binom{n}{n - 12}\).

### Step-by-step Solution:

1. **Set up the equation**:
Given:
\[ \binom{n}{12} = \binom{n}{8} \]

2. **Apply the property of combinations**:
From the property mentioned above, we can rewrite \(\binom{n}{12}\) as:
\[ \binom{n}{12} = \binom{n}{n - 12} \]
Thus, we can equate:
\[ \binom{n}{n - 12} = \binom{n}{8} \]

3. **Set the two expressions equal**:
Since both sides are equal, we can set the parameters equal to each other:
\[ n - 12 = 8 \]

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

5. **Verification**:
To confirm our solution, we can check if \(\binom{20}{12} = \binom{20}{8}\):
- Calculate \(\binom{20}{12}\):
\[ \binom{20}{12} = \frac{20!}{12! \cdot (20 - 12)!} = \frac{20!}{12! \cdot 8!} \]
- Calculate \(\binom{20}{8}\):
\[ \binom{20}{8} = \frac{20!}{8! \cdot (20 - 8)!} = \frac{20!}{8! \cdot 12!} \]
- Both expressions are equal, confirming that \(n = 20\) is correct.

### Final Answer:
\[ n = 20 \]