If a variable takes value 0,1,2,3,....,n with frequencies 1,C(n,1),C(n,2),C(n,3),...,C(n,n) respectively, then the arithmetic mean is

Options

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

Answer

To find the arithmetic mean of the variable that takes values \(0, 1, 2, 3, \ldots, n\) with corresponding frequencies \(1, C(n, 1), C(n, 2), C(n, 3), \ldots, C(n, n)\), we can follow these steps:

### Step 1: Identify the values and frequencies
The values of the variable are \(x = 0, 1, 2, 3, \ldots, n\) and their corresponding frequencies are:
- Frequency of \(0\) is \(1\)
- Frequency of \(1\) is \(C(n, 1)\)
- Frequency of \(2\) is \(C(n, 2)\)
- Frequency of \(3\) is \(C(n, 3)\)
- ...
- Frequency of \(n\) is \(C(n, n)\)

### Step 2: Calculate the total frequency
The total frequency \(N\) can be calculated as:
\[ N = 1 + C(n, 1) + C(n, 2) + C(n, 3) + \ldots + C(n, n) \]
Using the binomial theorem, we know that:
\[ 1 + C(n, 1) + C(n, 2) + \ldots + C(n, n) = 2^n \]
Thus, \(N = 2^n\).

### Step 3: Calculate the sum of \(x \cdot f\)
Now, we need to calculate the sum of \(x \cdot f\):
\[ \sum (x \cdot f) = 0 \cdot 1 + 1 \cdot C(n, 1) + 2 \cdot C(n, 2) + 3 \cdot C(n, 3) + \ldots + n \cdot C(n, n) \]
This can be expressed as:
\[ \sum_{r=1}^{n} r \cdot C(n, r) \]

### Step 4: Use the identity for \(r \cdot C(n, r)\)
We can use the identity:
\[ r \cdot C(n, r) = n \cdot C(n-1, r-1) \]
Thus, we can rewrite the sum:
\[ \sum_{r=1}^{n} r \cdot C(n, r) = n \sum_{r=1}^{n} C(n-1, r-1) = n \cdot 2^{n-1} \]
This is because \(\sum_{r=0}^{n-1} C(n-1, r) = 2^{n-1}\).

### Step 5: Calculate the arithmetic mean
Now we can find the arithmetic mean \( \bar{x} \):
\[ \bar{x} = \frac{\sum (x \cdot f)}{N} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2} \]

### Final Result
Thus, the arithmetic mean is:
\[ \bar{x} = \frac{n}{2} \]