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

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(C) Let $S$ be a set with $N = 2n + 1$ elements. We want to find the number of subsets with at most $n$ elements.This is given by the sum: $\sum_{r=0}

Vedclass · Accepted Answer

$2^{2n}$

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(C) Let $S$ be a set with $N = 2n + 1$ elements. We want to find the number of subsets with at most $n$ elements.This is given by the sum: $\sum_{r=0}^{n} {}^{2n+1}C_r = {}^{2n+1}C_0 + {}^{2n+1}C_1 + \dots + {}^{2n+1}C_n$.We know the property of binomial coefficients: ${}^{N}C_r = {}^{N}C_{N-r}$.Thus,the total number of subsets is $\sum_{r=0}^{2n+1} {}^{2n+1}C_r = 2^{2n+1}$.Since ${}^{2n+1}C_0 + {}^{2n+1}C_1 + \dots + {}^{2n+1}C_n + {}^{2n+1}C_{n+1} + \dots + {}^{2n+1}C_{2n+1} = 2^{2n+1}$,and by symmetry,the sum of the first $(n+1)$ terms equals the sum of the last $(n+1)$ terms:$2 	imes \sum_{r=0}^{n} {}^{2n+1}C_r = 2^{2n+1}$.Therefore,$\sum_{r=0}^{n} {}^{2n+1}C_r = \frac{2^{2n+1}}{2} = 2^{2n}$.

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

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