Question

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

• A. $2^n$
• B. $2^{n+1}$
• C. $2^{n-1}$
• D. $2^{2n}$

Answer 1

The correct option is A

$2^n$

Explanation for the correct option:

Step 1. Subsets of this set which contain at most n elements is:

$N={}^{2n+1}C_0+{}^{2n+1}C_1+{}^{2n+1}C_2+\dots +{}^{2n+1}C_n$

Step 2. Multiply both sides by 2:

$2N=2({}^{2n+1}C_0+{}^{2n+1}C_1+{}^{2n+1}C_2+\dots +{}^{2n+1}C_n)$

$=({}^{2n+1}C_0+{}^{2n+1}C_{2n+1})+({}^{2n+1}C_1+{}^{2n+1}C_{2n})+\dots +({}^{2n+1}C_n+{}^{2n+1}C_{n+1})$

$={}^{2n+1}C_0+{}^{2n+1}C_1+{}^{2n+1}C_2+\dots +{}^{2n+1}C_{2n+1}$ ….$\left[\mathbf{\because }{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}\mathbf{}{}^{\mathbf{n}}\mathit{C}_{\mathbf{n}\mathbf{–}\mathbf{r}}\mathbf{}\right]$

$=2^{2n+1}$

$\Rightarrow N={\mathbf{2}}^{\mathbf{2}\mathbf{n}}$

Hence, Option ‘D’ is Correct.