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### NCAA Tournament Possibilities

```
Date: 03/14/2001 at 15:28:45
Subject: NCAA tournament possibilities

If I had an NCAA Tournament office pool, where each person fills out
the brackets by selecting who he or she thinks will win each game,
how many possible combinations of choices are there?
```

```
Date: 03/14/2001 at 17:12:28
From: Doctor Paul
Subject: Re: NCAA tournament possibilities

A great question at a wonderful time of the year!

First notice that the 64 teams play 63 total games: 32 games in the
first round, 16 in the second round, 8 in the 3rd round, 4 in the
regional finals, 2 in the final four, and then the national
championship game.

32+16+8+4+2+1= 63

Now let's answer an easier question.

If there were four teams, and they played three games, how many
different ways would there be to fill out a bracket?  You can write
them down. There are only eight of them. Where do that eight come
from?  Well, there are three games, and you have two possible choices
for each game. Hence, 2^3 = 8 possibilities.

Now back to the real tournament. Since there are 63 games to be
played, and you have two choices at each stage in your bracket, there
are 2^63 different ways to fill out the bracket.

2^63 = 9,223,372,036,854,775,808

That's more than nine quintillion possibilities.

Perhaps now you can see why CNNSI is so willing to give away ten
million dollars to the person that fills out a perfect bracket. It's
just not likely to happen.

Even if every person in America filled one out (300 million people),
the probability of someone winning is:

300,000,000
----------- = .00000000003253
2^63

I don't think anyone is going to fill out a perfect bracket any time
soon.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 03/14/2006 at 12:24:36
From: Doctor Ian
Subject: NCAA tournament possibilities

A lot of people have written to us, asking whether this really takes into
account the structure of the tournament, the fact that once a team is
eliminated it can't play again, and so on.  A thought experiment can help
show why 2^63 is the correct answer.

Imagine that you're going to fill in a bracket by flipping a coin.  Each
time you get heads, you fill in the first team (the one above, or to the
left).  If you get tails, you fill in the second team.

Each possible sequence of 63 coin flips corresponds to a unique way of
filling in a bracket.  So the number of ways to fill in a bracket has
to be the same as the number of unique sequences of 63 coin flips, which
is 2^63.

I hope this helps clear up any confusion.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/

```
Associated Topics:
High School Permutations and Combinations

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