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Step-by-Step Solution
Step 1: Understand the Data and Frequencies
The given data points for the variable x are:
0, 2, 4, 8, …, 2^n .
Their corresponding frequencies are:
{}^nC_0 , {}^nC_1 , {}^nC_2 , …, {}^nC_n .
Step 2: Write Down the Formula for the Mean
The mean of a set of values x_i with frequencies f_i is given by:
\text{Mean} = \frac{\sum(x_i \cdot f_i)}{\sum f_i}.
In this problem:
x_i takes the values 0, 2, 4, \dots, 2^n
f_i takes the values {}^nC_0, {}^nC_1, \dots, {}^nC_n
Step 3: Express the Numerator of the Mean Using Binomial Expansion
To find \sum (x_i \cdot f_i) , observe that:
\sum_{k=0}^{n} (2^k\, {}^nC_k) = {}^nC_1 \cdot 2^1 + {}^nC_2 \cdot 2^2 + \dots + {}^nC_n \cdot 2^n.
We can use the binomial theorem for (1 + x)^n here:
(1 + x)^n = \sum_{k=0}^{n} {}^nC_k \, x^k.
By substituting x=2 , we get:
(1 + 2)^n = 3^n = \sum_{k=0}^{n} {}^nC_k \, 2^k.
However, note that the term k=0 in the summation is {}^nC_0 \cdot 2^0 = 1 . So,
\sum_{k=1}^{n} {}^nC_k \, 2^k = 3^n - {}^nC_0 \cdot 2^0 = 3^n - 1.
Thus, the numerator of the mean is 3^n - 1 .
Step 4: Express the Denominator of the Mean Using Binomial Expansion
The sum of frequencies is:
\sum_{k=0}^{n} {}^nC_k.
By the binomial theorem with x=1 :
(1 + 1)^n = 2^n = \sum_{k=0}^{n} {}^nC_k.
Hence, the denominator of the mean is 2^n .
Step 5: Calculate the Mean and Equate to the Given Value
Therefore, the mean is:
\text{Mean} = \frac{3^n - 1}{2^n}.
According to the question, the mean is also given as:
\frac{728}{2^n}.
Equating these two expressions for the mean:
\frac{3^n - 1}{2^n} = \frac{728}{2^n}.
This simplifies to:
3^n - 1 = 728 \quad \Rightarrow \quad 3^n = 729.
Step 6: Solve for n
Since 729 = 3^6 , it follows that:
n = 6.
Final Answer
The value of n is 6.
