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Pascal's triangle





Pascal's triangle
A triangular pattern of numbers in which each number is equal to the sum of the two numbers immediately above it. Although named after Blaise Pascal, who studied it, this arithmetic triangle has been known about since at least the 12th century and has a variety of other names. In Italy it is called Tartaglia's triangle (after Niccoló Tartaglia) and in many parts of Asia it is referred to as Yang Hui's triangle. Yang Hui was a minor Chinese official who wrote two books, dated 1261 and 1275, which use decimal fractions (long before they appeared in the West) and contain one of the earliest accounts of the triangle; at about the same time, Omar Khayyaam also wrote about it. The Chinese triangle appears again in 1303 on the front of Chu Shi-Chieh's Ssu Yuan YŁ Chien (Precious Mirror of the Four Elements), a book in which Chu says the triangle was known in China more than two centuries before his time.

The numbers in Pascal's triangle give the number of ways of picking r unordered outcomes from n possibilities. This is equivalent to saying that the numbers in each row are the binomial coefficients in the expansion of (x + y)n:

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

and so on. In addition, the shallow diagonals of the triangle sum to give the numbers in the Fibonacci sequence.

Pascal's triangle also has the following properties:
  • Each number is the sum of the two numbers standing above it to the left and right; e.g. 10 = 4 + 6.

  • Each number is equal to the sum of all numbers in the left or right diagonal, beginning with the number immediately above to the left or right, and proceeding upwards, e.g. 15 = 5 + 4 + 3 + 2 + 1 and 15 = 10 + 4 + 1.

  • Each diagonal is an arithmetic sequence; e.g.
       1st diagonal: 1, 1, 1, 1, 1, ... (arithmetic sequence of zero order)
       2nd diagonal: 1, 2, 3, 4, 5, ... (arithmetic sequence of 1st order)
       3rd diagonal: 1, 3, 6, 10, 15, ... (arithemtic sequence of 2nd order)
       4th diagonal: 1, 4, 10, 20, ... (arithmetic sequence of 3rd order)
       and so on.

Related entry

   • combinatorics