A) \[\frac{{{2}^{n}}}{n+1}\]
B) \[\frac{{{2}^{n+1}}}{n(n+1)}\]
C) \[\frac{n+1}{2}\]
D) \[\frac{n}{2}\]
Correct Answer: D
Solution :
The required mean is \[\bar{x}=\frac{0.1+{{1.}^{n}}{{C}_{1}}+{{2.}^{n}}{{C}_{2}}+{{3.}^{n}}{{C}_{3}}+......+n{{.}^{n}}{{C}_{n}}}{1{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}+....{{+}^{n}}{{C}_{n}}}\] \[=\frac{\sum\limits_{r=0}^{n}{r.\,{{\,}^{n}}{{C}_{r}}}}{\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}}}=\frac{\sum\limits_{r=1}^{n}{r.\frac{n}{r}\,{{\,}^{n-1}}{{C}_{r-1}}}}{\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}}}\]= \[\frac{n\sum\limits_{r=1}^{n}{^{n-1}{{C}_{r-1}}}}{\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}}}\] \[=\frac{n{{.2}^{n-1}}}{{{2}^{n}}}\] \[=\frac{n}{2}\].
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