$2^{63}=9.22337\cdot10^{18}$
Now, since it has never happened in history that a 16-seed team has beaten a 1-seed team in the first round (a whopping 0-96 record), then the chances get a little better if we factor those games out,
$2^{59}=5.76461\cdot10^{17}$
To put that large number in perspective, if we assume there are 7,222,626,382 (found from http://www.worldometers.info/world-population/ at 10:38 AM on March 28th, 2014), then each person would have to fill out 79,813,176.23 brackets each to fulfill every possible outcome of March Madness. Again another large number, so lets put that number into perspective. If every person were to fill one bracket completely every second, then it would take 2.53086 years to created every possible bracket outcome for this year's March Madness.
We can now show the chances of getting a perfect bracket if each team has a 50-50 shot to win each game.
$.5^{63}=1.08420\cdot10^{-19}$
$=1.08420\cdot10^{-17}\%$
Again, we could take out the 1-seed versus 16-seed games again, so the chances get a bit better,
$.5^{59}=1.73472\cdot10^{-18}$
$=1.73472\cdot10^{-16}\%$
Again, the chances are VERY slim of getting a perfect bracket. These numbers can again be adjusted though since every game has statistics that go into them and each team has a certain percent chance of winning any single game. If we factor in the odds. If we never pick an upset, then the chances get much better. The biggest toss-up of a game is the 8-seed versus the 9-seed at 47% of the time the 8-seed winning. From there, the higher seeded team wins more consistently from in an almost linear fashion from the 7-seed to the 1-seed. In the second round, 7-seeds win 60%, 6-seeds win 67%, 5-seeds win 67%, 4-seeds win 79%, 3-seeds win 85%, 2-seeds win 96%, and as mentioned above, 1-seeds win 100% of the time (from http://statistics.about.com/od/Applications/a/March-Madness-Statistics.htm). Using these statistics, we can make a better estimate of what the chances would be if picking the higher percentage win,
$1.00\cdot.96\cdot.85\cdot.79\cdot.67\cdot.67\cdot.6\cdot.53=.092022$
$= 9.2022\%$
After only the round of 64, picking the seeds with the best probability still only leaves 9.2022% for each region of having a perfect bracket. Factoring in that there are four different regions makes the percentage,
$.092002^4=.0071709$
$.71709\%$
Therefore, after only one round, and picking the higher seeded teams every time, the percentage of having a perfect bracket is only .71709%. In the next round, we are only left with the 1-seed versus 9-seed, 2-seed versus 7-seed, 3-seed versus 6-seed, and 4-seed versus 5-seed in each of the regions. In the third round, the 1-seeds win 91% against the 9-seed, the 2-seeds win 75% against the 7-seed, the 3-seeds win 57% against the 6-seeds, and the 4-seeds win 54% against the 5-seeds (from http://www.betfirm.com/seeds-national-championship-odds/). That means that the chances of getting the next round correct is,
$.91\cdot.75\cdot.57\cdot.54=.2100735$
$=21.00735\%$
Now, that is only for one region, again if all four regions are factored in, then
$.2100735^4=.001948$
$=.1948\%$
As you can see, as the rounds progress, the chances get a little bit better that the region can be gotten perfect. The next round would then be the 1-seed versus the 4-seed and the 2-seed versus the 3-seed. The 1-seed wins 58% against the 4-seed (from http://bluedevilnation.net/2014/03/path-final-four/) and the 2-seed wins 59% against the 3-seed (from http://www.sportingcharts.com/articles/nba/2014-ncaa-march-madness-cheat-sheet.aspx).
$.58\cdot.59=.3422$
$=34.22\%$
This is for each individual region, for all four regions it would be
$.3422^4=.01371$
$=1.371\%$
Finally, the last round before the final four would be the 1-seed versus the 2-seed. Picking the 1-seed has a 55% chance of being correct. Therefore,
$.55^4=.09151$
$=9.150625\%$
Since the last three games are then 1-seed versus 1-seed, then the chances are 50-50 for each game, thus
$.5^3=.125$
$=12.5\%$
To figure out the whole percentage of getting the bracket perfect, you must multiply all the percentages together. Therefore,
$.0071709*.0019475*.091506*.013713*.125=2.1905\cdot10^{-9}$
$=2.1905\cdot10^{-7}\%$
$2.1905\cdot10^{-7}\%-1.73472\cdot10^{-16}\%=2.1905\cdot10^{-7}\%$
Therefore, with going by the percentages, your chances are much better. Your chances are about 10,000,000,000 times better than if each game was a 50-50 shot.
If this sort of stuff interests you more, there are many good sites to look at in order to get more information. I have put where I put my references, but other places for information you can look at,
http://fivethirtyeight.com/features/nate-silvers-ncaa-basketball-predictions/
http://bracketodds.cs.illinois.edu/index.html
http://outsidethehashes.com/?p=471
http://www.sportingcharts.com/articles/nba/how-does-seeding-affect-success-in-the-ncaa-mens-basketball-tournament.aspx
http://www.esquire.com/blogs/news/2014-march-madness-bracket-stats
