Question

If a set $A$ has $n$ elements, then the total number of subsets of $A$ is

• A. $n$
• B. $n^2$
• C. $2^n$
• D. $2n$

Answer 1

The correct option is C

$2^n$

A set $B$ is a subset of another set $A$ if all elements of the set $B$ are elements of the set $A$ and represented by $B\subseteq A$.
Here,it is given that set $A$ contains $n$ elements.
We need to find all possible subsets of set $A$ which can have $0,1,2,3,...,n$ elements of set $A$.
As we know ${}^n{C}_r$ denotes the number of ways to select $r$ items from $n$ distinct items.

Hence the required selection can be done as follows:
$\therefore$ Number of subsets of $A$
$={}^n{C}_0{+}^n{C}_1+\cdots {+}^n{C}_n$
$=2^n[\because \text{sum of binomial coefficients}]$