Techniques for summing ratio of binomial coefficients
Solution 1:
There is a technique known as Gosper's algorithm and another technique which is known as Zeilberger's algorithm. The two algorithms tackle these kinds of problems, if they succeed they will give a closed form formulas for the finite sum.
Solution 2:
Take a look at Petkovsek and Wilf's book "A=B", they cover techniques for such hypergeometric sums in full detail. There are packages for their algorithms (which normally are much to messy for hand computation) for leading computer algebra packages.