Find the number of functions $h:\{1,2,3,\ldots ,2n\}\to \{-1,1\}$ such that $\sum_{j=1}^{2n}h(j)>0$ . 
Fix $n\in \Bbb N$.
  Find the number of functions $h:\{1,2,3,\ldots ,2n\}\to \{-1,1\}$
  such that $$\sum_{j=1}^{2n}h(j)>0$$

My try:
Let $A=\{x:h(x)=1\}$
and $B=\{x:h(x)=-1\}$
In order for $\sum_{j=1}^{2n}h(j)>0$  we should have $|A|>|B|$
where $|A|$ denotes cardinality of set $A$.
Now Cardinality of $B$ can have values from the set  $\{0,1,2,3,\ldots ,n-1\}$
When we fix $B$ 's cardinality the cardinality of $A$ gets fixed too.
Thus if $B$ has cardinality $j$ then the $j$ elements can be chosen from  $\{1,2,3,\ldots ,2n\}$ in $\binom{2n}{j}$ ways.
Thus the total number of functions become
$$\sum_{j=0}^{n-1}\binom{2n}{j}$$
Is my solution correct?Kindly check 
 A: It's correct, but we can do better - a closed form in which we're not summing an increasing number of terms.
Exactly half of the functions with nonzero sum have positive sum, by symmetry. There are $2^{2n}$ total functions. There are $\binom{2n}{n}$ functions with sum zero, by your argument. Therefore, our answer is
$$\frac12\left(2^{2n}-\binom{2n}{n}\right)$$
A: You can also arrive at jmerry's closed form answer directly from your answer. You got:
$$\sum_{j=0}^{n-1}\binom{2n}{j}$$
Now, notice that
$$\sum_{j=0}^{n-1}\binom{2n}{j} = \sum_{j=0}^{n-1} \binom{2n}{2n-j} = \sum_{j=n+1}^{2n} \binom{2n}{j}.$$
Therefore, the answer can be written
\begin{align*}
\sum_{j=0}^{n-1}\binom{2n}{j}
&= \frac12\;\left({\displaystyle\sum_{j=0}^{n-1}\binom{2n}{j} + \sum_{j=0}^{n-1}\binom{2n}{j}}\right) \\
&= \frac12\;\left({\displaystyle\sum_{j=0}^{n-1}\binom{2n}{j} + \sum_{j=n+1}^{2n} \binom{2n}{j}}\right) \\
&= \frac12\;\left({\displaystyle\left\{\sum_{j=0}^{2n}\binom{2n}{j}\right\} - \binom{2n}{n}}\right) \\
&= \frac12\;\displaystyle\left(2^{2n} - \binom{2n}{n}\right).
\end{align*}
