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Pascal's Triangle Tidbits


Date: 5 Apr 1995 16:37:06 -0400
From: Library Media Center
Subject: Pascal's Triangle

Dear Dr. Math:

    My friend and I are doing a math fair project on Pascal's Triangle.  
We would like to know if you could send us some information about it.  
We are writing to you because our library is very limited on the subject.  
We would really appreciate it if you could send us the information as 
soon as possible.  

                                               Sincerely,
                                               Thelma and Ana Maria 10th

P.S. In your subject line please put our name and grade so they can give 
it directly to us.  Our E mail number is acig@acig.k12.ec 


Date: 7 Apr 1995 11:36:14 -0400
From: Dr. Ethan
Subject: to Thelma and Ana 10th-Pascal's triangle

Hey Folks,

Pascal's triangle is pretty neat and has lots of uses.  I only know 
the first little bit.  I will start by telling you my favorite uses; then 
I'll give you some books that you might want to use.

My favorite uses of Pascal's triangle is in binomial expansions.  For
instance, what if you wanted the fifth term of (x+y)^4? That would be a 
pretty nasty calculation unless you knew this neat thing about Pascal's 
triangle:  the coefficients of a binomial expansion are associated with 
Pascal's triangle in a really neat way.  

(x+y)^4  = 1 x^4  + 4 x^3 y  + 6 x^2 y^2 + 4 x y^3 + 1 y^4 

You may notice that the coefficients are the numbers in the fourth row of
Pascal's triangle.  Pretty neat huh.

Another neat little tidbit:  the triangular numbers and the Fibonacci 
numbers can be found in the Pascal's triangle.  The triangle numbers are 
easier to find.  Starting with the third one on the left side go down to 
your right and you get 1,3,6,10 etc. These are the triangular numbers.

The Fibonacci numbers are harder to find.  To get them you need to go up 
at an angle.  The numbers you get are 1, 1, 1+1, 1+2, 1+3+1, 1+4+3, 
1+5+6+1.  These are the Fibonacci numbers.  If you don't see how I am 
getting these numbers, please right back to me and I will try to 
explain these a little better.

Hope all this helps,

Ethan Doctor On Call
    
Associated Topics:
High School Discrete Mathematics

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