Pascal's Triangle (Reading Level 6)

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Pascal’s Triangle is a mathematical triangular array.[1] It is named after French mathematician Blaise Pascal, but it was used in China at least 300 years before Pascal was born.

"Blaise Pascal"
caption Blaise Pascal

Pascal’s triangle looks like a triangle with 1 number at the top and then each row having one more number—so, 2 numbers in the second row, 3 numbers in the third row, etc. The important thing is the numbers that go in each row.

The triangle can be used to find out the coefficients of the terms of a binomial raised to a positive power. For example, <math>(x + 1)^4</math> would not be much fun to calculate by hand, but with Pascal’s Triangle, we can just look at the fifth row and see that this expression will be <math>x^4 + 4x^3 + 6x^2 + 4x + 1</math>.

"Pascal's Triangle"
caption Pascal’s Triangle, for exponents 0 to 5

Making the Triangle

The triangle can be made as follows:

  1. In the first row, write only the number 1.
  2. In the second row, write the number 1 twice.
  3. To get the numbers of following rows, add the two numbers to the left and right in the row above. If there is no number to the left or to the right, then pretend that that number is zero. That is why both numbers in the second row are 1—we pretend that the first row is <math>0 1 0</math>.
  4. Continue like this for as many rows as you need.



  1. array: a bunch of numbers aligned in rows or columns