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

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

### Step-by-Step Solution:

1. **Apply the Property of Combinations:**
We know that \( \binom{n}{r} = \binom{n}{n-r} \). In this case, we can set \( r = 12 \) and \( n - r = 8 \).

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

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

4. **Verification:**
To verify, we can check if \( \binom{20}{12} \) is indeed equal to \( \binom{20}{8} \):
\[ \binom{20}{12} = \binom{20}{20-12} = \binom{20}{8} \]
This confirms that our solution is correct.

### Final Answer:
Thus, the value of \( n \) is \( 20 \).

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