Question

A set contains $n$ elements. The power set contains

• A. $n$ elements
• B. $2^n$ elements
• C. $n^2$ elements
• D. None of these

Answer 1

The correct option is B $2^n$ elements

Let $A$ be a finite set containing $n$ elements let $0\le r\le n$.

Consider those subset of $A$ that have $r$ elements each.

We know that the number of ways in which $r$ elements can be chosen out of $n$ elements is ${}^n{C}_r$.

Therefore, the number of subset of $A$ having $r$ elements each is ${}^n{C}_r$.

Therefore, the total number of subset of $A$ i.e., the total number of elements in power set of $A$
$={}^n{C}_0+{}^n{C}_1+{}^n{C}_2+...+{}^n{C}_n$
$=(1+1{)}^n=2^n$.