A set contains (2n + 1) elements. The number | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $S$ be a set with $N = 2n + 1$ elements. We want to find the number of subsets with at most $n$ elements,which is given by the sum: $S = \sum_

Vedclass · Accepted Answer

$2^{2n}$

Vedclass · Answer

(B) Let $S$ be a set with $N = 2n + 1$ elements. We want to find the number of subsets with at most $n$ elements,which is given by the sum: $S = \sum_{r=0}^{n} \binom{2n+1}{r} = \binom{2n+1}{0} + \binom{2n+1}{1} + \dots + \binom{2n+1}{n}$. We know that the total number of subsets is $\sum_{r=0}^{2n+1} \binom{2n+1}{r} = 2^{2n+1}$. By the property of combinations,$\binom{N}{r} = \binom{N}{N-r}$. Thus,$\binom{2n+1}{0} = \binom{2n+1}{2n+1}$,$\binom{2n+1}{1} = \binom{2n+1}{2n}$,...,$\binom{2n+1}{n} = \binom{2n+1}{n+1}$. Let $X = \sum_{r=0}^{n} \binom{2n+1}{r}$. Then $X = \binom{2n+1}{0} + \dots + \binom{2n+1}{n}$. Also,$X = \binom{2n+1}{2n+1} + \dots + \binom{2n+1}{n+1}$. Adding these two expressions: $2X = \sum_{r=0}^{2n+1} \binom{2n+1}{r} = 2^{2n+1}$. Therefore,$X = \frac{2^{2n+1}}{2} = 2^{2n}$.

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

Similar Questions

$A$ man has $10$ friends. In how many ways can he invite one or more of them to a party?

From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

There are $12$ volleyball players in total in a college,out of which a team of $9$ players is to be formed. If the captain always remains the same,in how many ways can the team be formed?

Vedclass Test Series

Exam Paper Generator

Online Exam Module