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An Explanation of Pascal's Triangle


Date: 27 Feb 1995 22:57:04 -0500
From: Anonymous
Subject: Probability and Pascal's Triangle

My 10th grade daughter is having trouble finding a simple explanation 
of Pascal's Triangle and its application.  So far we've discovered that, 
in lieu of prozac, Pascal turned to math and came up with the 
arithmetical triangle as an aid for his gambling buddies.  But that's 
where our information runs out.  Any help would be gratefully appreciated.

Erin and Dale in Helena, Montana


Date: 28 Feb 1995 14:19:21 -0500
From: Dr. Ken
Subject: Re: Probability and Pascal's Triangle

Hello there!

Well, since I'm not quite sure how much you know about Pascal's Triangle
(for instance, you seem to know that it is connected to the study of
probability), I'll start pretty much from the beginning.

The way I see it, Pascal's Triangle is kind of a collection of neat things
in mathematics.  The way you construct it follows:

                                  1
                                1   1
                              1   2   1
                            1   3   3   1
                          1   4   6   4   1
                        1   5  10   10  5   1
                      1   6  15  20   15  6   1
                    1   7  21  35   35  21  7   1
                                  .
                                  .
                                  .

You start out with the top two rows: 1, and 1 1.  Then to construct each
entry in the next row, you look at the two entries above it (i.e. the one
above it and to the right, and the one above it and to the left).  At the
beginning and the end of each row, when there's only one number above, 
put a 1.  You might even think of this rule (for placing the 1's) as included 
in the first rule: for instance, to get the first 1 in any line, you add up the
number above and to the left (since there is no number there, pretend it's
zero) and the number above and to the right (1), and get a sum of 1.

When people talk about an entry in Pascal's Triangle, they usually give a
row number and a place in that row, beginning with row zero and place 
zero.  For instance, the number 20 appears in row 6, place 3.

That's how you construct Pascal's Triangle.  An interactive version where 
you can specify the number of rows you want to see can be found at

  http://mathforum.org/dr.cgi/pascal.cgi   

But Pascal's Triangle is more than just a big triangle of numbers. There 
are two huge areas where Pascal's Triangle rears its head, in Algebra 
and in Probability/Combinatorics.  

First let's look at the Algebra version.

Let's say you have the polynomial 1+x, and you want to raise it to some
powers, like 1,2,3,4,5,....  If you make a chart of what you get when you 
do these power-raisings, you'll get something like this:

(x+1)^0   =                           1
(x+1)^1   =                      1    +    x
(x+1)^2   =                 1    +   2x    +    x^2
(x+1)^3   =             1   +   3x    +   3x^2  +    x^3
(x+1)^4   =         1   +  4x    +   6x^2  +   4x^3  +    x^4
(x+1)^5   =     1   +  5x   +  10x^2  +  10x^3  +   5x^4  +    x^5  .....


If you just look at the coefficients of the polynomials that you get, voila!
Pascal's Triangle!  Because of this connection, the entries in Pascal's
Triangle are called the _binomial_coefficients_.  They are usually written
in parentheses, with one number on top of the other, for instance 

20 =  (6)   <--- note: that should be one big set of 
      (3)              parentheses, not two small ones.

I don't think it's standard notation, but when I write binomial coefficients
in a text document like this, I usually write them [6:3].

The other main area where Pascal's Triangle shows up is in Probability.
Let's say you have five hats on a rack, and you want to know how many
different ways you can pick two of them to wear.  It doesn't matter 
to you which hat is on top, it just matters which two hats you pick. 
So this problem amounts to the question "how many different ways can you 
choose two objects from a set of five objects?" The answer? It's the 
number in the second place in the fifth row, i.e. 10. (Remember that the 
first number is in place zero.) 

Because of this choosing property, the binomial coefficient [6:3] is 
usually read "six choose three."  If you want to find out the probability 
of choosing one particular combination of two hats, that probability is 1/10.

There's a pretty simple formula for figuring out the binomial coefficients.
It's 
            n!
[n:k] = --------
        k! (n-k)!
                       6 * 5 * 4 * 3 * 2 * 1
For example, [6:3] =  ------------------------  =  20. 
                       3 * 2 * 1 * 3 * 2 * 1


That's a basic introduction to Pascal's Triangle.  It certainly also shows
up in lots of other places (for example, the triangular numbers are in
there, if you know what those are), but I think it would be too much for 
me to go into those right now.  Thanks for the question!

-Ken "Dr." Math
    
Associated Topics:
High School Probability

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