If `C(n,8)= C(n, 6)`, find C(n, 2)

Options

A. 91
B. 81
C. 61
D. 71

Answer

To solve the problem where \( C(n, 8) = C(n, 6) \) and find \( C(n, 2) \), we can follow these steps:

### Step 1: Use the property of combinations
We know that \( C(n, k) = C(n, n-k) \). Therefore, we can rewrite \( C(n, 6) \) as:
\[ C(n, 6) = C(n, n-6) \]
Given that \( C(n, 8) = C(n, 6) \), we can also express \( C(n, 8) \) as:
\[ C(n, 8) = C(n, n-8) \]
Thus, we have:
\[ C(n, n-8) = C(n, n-6) \]
This implies that:
\[ n - 8 = n - 6 \quad \text{or} \quad n - 8 = 6 \]

### Step 2: Solve for \( n \)
From the equation \( n - 8 = 6 \):
\[ n = 6 + 8 = 14 \]

### Step 3: Find \( C(n, 2) \)
Now that we have \( n = 14 \), we can find \( C(14, 2) \):
\[ C(14, 2) = \frac{14!}{2!(14-2)!} = \frac{14!}{2! \cdot 12!} \]
This simplifies to:
\[ C(14, 2) = \frac{14 \times 13}{2 \times 1} = \frac{182}{2} = 91 \]

### Conclusion
Thus, the value of \( C(n, 2) \) is \( 91 \).