Question

How many onto (or surjective) functions are there from an $n-$ element $(n\ge 2)$ set to a 2-element set?

• A. $2^n$
• B. $2^n-1$
• C. $2^n-2$
• D. $2(2^n-2)$

Answer 1

The correct option is C $2^n-2$

Let $n=2$. There are only 2 onto functions as shown below:

For, $n=2$
Option (a) $2^n=2^2=4$
Option (b) $2^n-1=2^2-1=3$
Option (c) $2^n-2=2^2-2=2$
Option (d) $2(2^n-2)=2(2^2-2)=4$
So only option (c) gives correct answer.
Alternate method:
The number of onto functions from a set $A$ with $m$ elements to set $B$ with $n$ elements where $n<m$ is given by
$n^m{-}^n{C}_1(n-1{)}^m{+}^n{C}_2(n-2{)}^m\dots \dots {+}^n{C}_{n-1}{1}^m$
Here, $m=n=2$
So number of onto functions $=2^n{-}^2{C}_1{1}^m$
$=2^n-2$
which is choice (c).