SATHEE: Permutations And Combinations Question 191

Permutations And Combinations Question 191

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} $ .

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Permutations And Combinations Question 191

Question: A set contains $ (2n+1) $ elements. The number of sub-sets of the set which contain at most $ n $ elements is

Options:

Answer:

Solution:

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