The arithmetic mean of the following frequency distribution:
`{:("Variate(X)" ,:,0,1,2,3, ..., n),("Frequency" (f),:,""^(n)C_(0),""^(n)C_(1),""^(n)C_(2),""^(n)C_(3),...,""^(n)C_(n)):}` is

Options

A. `(2^(n))/(n)`
B. `(2^(n))/(n+1)`
C. `(n)/(2)`
D. `(2)/(n)`

Explanation

To find the arithmetic mean of the given frequency distribution, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variate and Frequency**:
- The variate \( X \) is given as \( 0, 1, 2, 3, \ldots, n \).
- The frequency \( f \) is given as \( \binom{n}{0}, \binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n} \).

2. **Write the Formula for Arithmetic Mean**:
- The arithmetic mean \( \bar{X} \) is calculated using the formula:
\[ \bar{X} = \frac{\Sigma (X \cdot f)}{\Sigma f} \]
- Here, \( \Sigma (X \cdot f) \) is the sum of the product of each variate and its corresponding frequency, and \( \Sigma f \) is the total frequency.

3. **Calculate \( \Sigma f \)**:
- The total frequency \( \Sigma f \) is:
\[ \Sigma f = \sum_{r=0}^{n} \binom{n}{r} = 2^n \]
- This is based on the binomial theorem, which states that the sum of the binomial coefficients equals \( 2^n \).

4. **Calculate \( \Sigma (X \cdot f) \)**:
- We need to calculate \( \Sigma (X \cdot f) \):
\[ \Sigma (X \cdot f) = 0 \cdot \binom{n}{0} + 1 \cdot \binom{n}{1} + 2 \cdot \binom{n}{2} + \ldots + n \cdot \binom{n}{n} \]
- This can be expressed as:
\[ \Sigma (X \cdot f) = \sum_{r=0}^{n} r \cdot \binom{n}{r} \]
- Using the property of binomial coefficients, we know:
\[ r \cdot \binom{n}{r} = n \cdot \binom{n-1}{r-1} \]
- Therefore:
\[ \Sigma (X \cdot f) = n \cdot \sum_{r=1}^{n} \binom{n-1}{r-1} = n \cdot 2^{n-1} \]

5. **Substitute into the Mean Formula**:
- Now substitute \( \Sigma (X \cdot f) \) and \( \Sigma f \) into the mean formula:
\[ \bar{X} = \frac{n \cdot 2^{n-1}}{2^n} \]

6. **Simplify the Expression**:
- Simplifying the expression gives:
\[ \bar{X} = \frac{n}{2} \]

### Final Answer:
Thus, the arithmetic mean of the given frequency distribution is:
\[ \bar{X} = \frac{n}{2} \]