Ok, good to know you have already investigated solving it analytically. I have to admit it's been too many years for me to remember whether this could be done easily by hand.
However, there is a simple way to solve this without simulating the process millions of times. You just need to think about the probability distribution associated with coin flips (hint: it's the most simple one of all). This will then lead you to the set of functions in R (or Excel for that matter) which do the probability calculations for you.