What is the maximum value of `(12)C_x` for any natural number x ?

Options

A. 6
B. 7
C. 8
D. 4

Answer

To find the maximum value of \( \binom{12}{x} \) for any natural number \( x \), 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 for the binomial coefficient is:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

### Step 2: Identify the Value of \( n \)
In this problem, we have \( n = 12 \). We need to find the maximum value of \( \binom{12}{x} \).

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

### Step 4: Calculate \( \frac{n}{2} \)
Since \( n = 12 \) (which is even), we calculate:
\[ \frac{12}{2} = 6 \]
Thus, the maximum value of \( \binom{12}{x} \) occurs when \( x = 6 \).

### Step 5: Conclusion
The maximum value of \( \binom{12}{x} \) for any natural number \( x \) is when \( x = 6 \).

### Final Answer
The maximum value of \( \binom{12}{x} \) occurs at \( x = 6 \).

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