For what value of x, `(14)C_x` is maximum ?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To find the value of \( x \) for which \( \binom{14}{x} \) is maximum, we can follow these steps:

### Step 1: Understand the Binomial Coefficient
The binomial coefficient \( \binom{n}{r} \) represents the number of ways to choose \( r \) elements from a set of \( n \) elements. The formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

### Step 2: Identify the Value of \( n \)
In this case, we have \( n = 14 \). We need to determine the value of \( x \) (which corresponds to \( r \)) that maximizes \( \binom{14}{x} \).

### Step 3: Determine the Maximum Value
The maximum value of \( \binom{n}{r} \) occurs at the middle term when \( n \) is even. For an even \( n \), the maximum occurs at \( r = \frac{n}{2} \).

### Step 4: Calculate \( \frac{n}{2} \)
Since \( n = 14 \):
\[ \frac{14}{2} = 7 \]

### Step 5: Conclusion
Thus, the value of \( x \) for which \( \binom{14}{x} \) is maximum is:
\[ \boxed{7} \]