There's a math problem in the book Discrete Mathematics with Applications(2nd Edition) by Susanna S. Epp is something like this:

How many onto functions are there from a set with four elements to a set with two elements?

Definition of onto function is:

Let $\displaystyle F$ be a function from a set $\displaystyle X$ to a set $\displaystyle Y$. $\displaystyle F$ is onto(or surjective) if, and only if, given any element $\displaystyle y$ in

$\displaystyle Y$ it is possible to find an element $\displaystyle x$ in $\displaystyle X$ with the property that $\displaystyle y = F(x)$.

Symbolically:

$\displaystyle F:X\to Y\; is\; onto\; \Leftrightarrow \forall y\in Y,\; \exists x\in X\; such\; that\; F(x)\; =\; y.$

Now back to the math problem. I did this.

Let the element of the domain be called $\displaystyle M = \{a,b,c,d\}$ and the elements of the co-domain be called

$\displaystyle N = \{u,v\}$ Now there $\displaystyle \binom{4}{2} + \binom{4}{2} = 6 + 6 = 12$ ways to select 2 elements of $\displaystyle M$ and and associate with set $\displaystyle N$. I added twice because there are two elements in $\displaystyle N$ and 1 for each element of $\displaystyle N$

Also there are $\displaystyle \binom{4}{3}+\binom{4}{3} = 4 + 4 = 8$ ways to select 3 elements of $\displaystyle M$ and associate with set $\displaystyle N$. Again because there are two elements in $\displaystyle N$ and added twice.

So the total number of onto function from a set with four elements to a set with two elements is: $\displaystyle 12 + 8 = 20$. But the answer in the book is $\displaystyle 14$.

What am i doing wrong? Is it possible for someone to kindly look into the problem and figure out the error in my thinking?