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### Bingo

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Date: 2 Aug 1995 18:59:46 -0400
From: Anonymous
Subject: Bingo

Question: Is there any literature on the number of ways of winning at bingo?

Assume horizontal, vertical, and diagonal 5 in a row are required
to win.  How many possible different bingo cards are there?
```

```
Date: 3 Aug 1995 11:26:08 -0400
From: Dr. Ken
Subject: Re: bingo

Hello there!

One of the Doctors, Heather Mateyak, knew about the configuration of Bingo
cards, so she went ahead and solved the problem. Here it is.

The first row can contain the numbers 1-15 in any order, with no
duplicates.  The second number can contain 16-30, the third 31-45, the
fourth 46-60 and the fifth 61-75.  There's also a free space in the center
square that doesn't get a number.  So how can we fill in the spaces, given
that these are our choices for filling them in?  Well, there are fifteen ways
of filling in the first square, fourteen ways of filling in the next
one down (since we already chose the first square and can't duplicate the
number), thirteen for the next, then twelve, then 11.  So there are
15*14*13*12*11 different versions of the first column.  Likewise, there are
the same number of different versions of every other column except the middle
one, for which there are 15*14*13*12 different versions.  So to get the
total number of different possible Bingo cards, we multiply all these
together, and we get (15*14*13*12)^5 * 11^4, which is about 5.5 x 10^26.

-K
```
Associated Topics:
Middle School Puzzles

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