The arithmetic mean of `""^(n)C_(0),""^(n)C_(1),""^(n)C_(2), ..., ""^(n)C_(n)`, is

Options

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

Explanation

To find the arithmetic mean of the binomial coefficients \( \binom{n}{0}, \binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n} \), we can follow these steps:

### Step 1: Understand the formula for the arithmetic mean
The arithmetic mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]

### Step 2: Identify the observations
The observations in this case are the binomial coefficients:
\[ \binom{n}{0}, \binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n} \]

### Step 3: Calculate the total number of observations
The total number of observations is the number of binomial coefficients from \( \binom{n}{0} \) to \( \binom{n}{n} \). Since the coefficients start from 0 and go up to \( n \), there are \( n + 1 \) observations.

### Step 4: Calculate the sum of the observations
The sum of all the binomial coefficients from \( \binom{n}{0} \) to \( \binom{n}{n} \) can be calculated using the identity:
\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
Thus, the sum of observations is:
\[ \text{Sum} = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \ldots + \binom{n}{n} = 2^n \]

### Step 5: Substitute into the mean formula
Now, we can substitute the sum and the total number of observations into the mean formula:
\[ \text{Mean} = \frac{2^n}{n + 1} \]

### Final Result
Thus, the arithmetic mean of \( \binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n} \) is:
\[ \text{Mean} = \frac{2^n}{n + 1} \]

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