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The Yanghui Triangle

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© Copyright 1999, Jim Loy

You may recognize the following triangle as Pascal's Triangle, named after Blaise Pascal. But Yanghui, a Chinese mathematician, discovered it 500 years before Pascal did. It may have been discovered earlier, in China. It is a triangular arrangement of integers like this:

            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
...

Each number is the sum of the two numbers diagonally above it. This triangle gives the coefficients of the expression (x+y)n:

(x+y)0=1
(x+y)1=x+y
(x+y)2=x2+2xy+y2
(x+y)3=x3+3x2y+3xy2+y3
(x+y)4=x4+4x3y+6x2y2+4xy3+y4
...

The triangle gives the same results as the Binomial theorem:

(x+y)n=xn+(n!/1!(n-1!))xn-1y+(n!/2!(n-2!))xn-2y2+...

Or:

same equation in a different form

Where combination notation=n!/r!(n-r!) is the number of combinations of n things taken r at a time.

See The Sierpinski Gasket. Actually, it is uncertain who discovered this triangle in China, but they beat Pascal to the mathematical punch. It was later discovered in India, also before Pascal.

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