Question: A set contains $ (2n+1) $ elements. The number of sub-sets of the set which contain at most $ n $ elements is
Options:
A) $ 2^{n} $
B) $ {2^{n+1}} $
C) $ {2^{n-1}} $
D) $ 2^{2n} $
Show Answer
Answer:
Correct Answer: D
Solution:
- The number of sub-sets of the set which contain at most n elements is $ ^{2n+1}C_0{{+}^{2n+1}}C_1+…..+{{}^{2n+1}}C _{n}=S $ (Say) Then $ 2S=2{{(}^{2n+1}}C_0+{{}^{2n+1}}C_1+…..+{{}^{2n+1}}C _{n}) $ = $ {{(}^{2n+1}}C_0+{{}^{2n+1}}{C _{2n+1}})+{{(}^{2n+1}}C_1+{{}^{2n+1}}C _{2n})+……..+{{(}^{2n+1}}C _{n}+{{}^{2n+1}}{C _{n+1}}) $ $ { \because {{}^{n}}C _{r}={{}^{n}}{C _{n-r}} } $ = $ ^{2n+1}C_0+{{}^{2n+1}}C_1+…….+{{}^{2n+1}}{C _{2n+1}}={2^{2n+1}} $
$ \Rightarrow $ $ S=2^{2n} $ .
BETA
