Question:
Consider the data on $x$ taking the values 0,2, $4,8, \ldots, 2^{\mathrm{n}}$ with frequencies ${ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots$ ${ }^{n} C_{n}$ respectively. If the mean of this data is
$\frac{728}{2^{\mathrm{n}}}$, then $\mathrm{n}$ is equal to
Solution:
Mean $=\frac{\sum x_{i} f_{i}}{\sum f_{i}}=\frac{\sum_{r=1}^{n} 2^{r}{ }^{n} C_{r}}{\sum_{r=0}^{n}{ }^{n} C_{r}}$
Mean $=\frac{(1+2)^{\mathrm{n}}-{ }^{\mathrm{n}} \mathrm{C}_{0}}{2^{\mathrm{n}}}=\frac{728}{2^{\mathrm{n}}}$
$\Rightarrow \frac{3^{\mathrm{n}}-1}{2^{\mathrm{n}}}=\frac{728}{2^{\mathrm{n}}}$
$\Rightarrow 3^{\mathrm{n}}=729 \Rightarrow \mathrm{n}=6$