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

If the number of subsets with $8$ elements from the set $A=\{a_1, a_2, a_3, \ldots, a_n\}$,$n \geq 8$,is five times the number of such subsets containing $a_4$,then $n=$

$A$ student has to answer $10$ out of $13$ questions in an examination,choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is:

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

In an election,$3$ members are to be elected from $6$ candidates. $A$ voter can cast their vote for any number of candidates but not more than the number of candidates to be elected. In how many ways can a voter cast their vote?

The number of ways in which one or more balls can be selected out of $10$ white,$9$ green,and $7$ blue balls is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module