IF `""^n C_12=""^nC_6` then `"^n C_2`=

Options

A. 82
B. 153
C. 206
D. 256

Answer

To solve the problem where \( \binom{n}{12} = \binom{n}{6} \) and we need to find \( \binom{n}{2} \), we can follow these steps:

### Step 1: Understand the relationship between combinations
We know that:
\[ \binom{n}{r} = \binom{n}{n-r} \]
This means that \( \binom{n}{12} = \binom{n}{n-12} \) and \( \binom{n}{6} = \binom{n}{n-6} \).

### Step 2: Set up the equation
Given \( \binom{n}{12} = \binom{n}{6} \), we can use the property from Step 1:
\[ \binom{n}{12} = \binom{n}{n-12} \]
Thus, we have:
\[ \binom{n}{n-12} = \binom{n}{6} \]
This implies:
\[ n - 12 = 6 \quad \text{or} \quad n - 6 = 12 \]

### Step 3: Solve for \( n \)
From the equation \( n - 12 = 6 \):
\[ n = 6 + 12 = 18 \]
From the equation \( n - 6 = 12 \):
\[ n = 12 + 6 = 18 \]
In both cases, we find that \( n = 18 \).

### Step 4: Calculate \( \binom{n}{2} \)
Now we need to find \( \binom{18}{2} \):
\[ \binom{18}{2} = \frac{18 \times 17}{2 \times 1} \]
Calculating this gives:
\[ \binom{18}{2} = \frac{306}{2} = 153 \]

### Final Answer
Thus, the value of \( \binom{n}{2} \) is:
\[ \boxed{153} \]