Pascal's Triangle and Powers of 11

Date: 02/21/99 at 23:01:18
From: Robert Chung
Subject: Pascal's Triangle and Powers of 11

Hi there.  I've been working on a project for my finite math class and 
was wondering if you could help me with a certain pattern found in 
Pascal's Triangle.  There seems to be a relation between the rows of 
Pascal's Triangle and the Powers of 11.  For example:

                                 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
                1   8   28  56   70   56  28  70  1

The first row (1) is 11^0       = 1
The second row (1 1) is 11^1    = 11
The third row is (1 2 1) 11^2   = 121
And so on.

However, the pattern becomes harder to see after the 6th row; i.e: 
1  5  10  10  5  1 becomes 11^5 = 161051

A funny way to work this out is:
1   5     10       10     5   1
1  (5 + 1)  (0 + 1)  (0 + 5)  1

The rest of the rows are harder to get. I was just wondering if there's 
a special way to find the pattern in each row of Pascal's Triangle, and 
to see whether a pattern can be found at the 12th row (1 11 55 165 330 
462 330 165 55 11 1).

Thanks a bunch.

-Rob-

Date: 02/21/99 at 23:13:53
From: Doctor Schwa
Subject: Re: Pascal's Triangle and Powers of 11

Indeed, Pascal's triangle gives the coefficients of (x+y)^n in general, 
so for example the 1 3 3 1 row tells you that 

    (x+y)^3 = x^3 + 3x^2 y + 3xy^2 + y^3

If you let x = 10, and y = 1, then you have

    11^3 = 10^3 + 3*10^2 * 1 + 3*10*1^2 + 1^3

and since all the powers of 1 are just 1, you get 1331.

For later rows, when you have numbers in the triangle that are bigger
than 10, they "carry" to the next row.  So, if you use the 12th row as
you have it, you can just add

1
11
 55
 165
  330
   462
    330
     165
       55
        11
          1

where the numbers are lined up so the rightmost digit moves over one
unit each time (because you have one fewer power of ten to multiply it
by).

I hope that helps clear things up for you!

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