Pascal's Triangle

Pronunciation: /pɑˈskɑlz ˈtraɪˌæŋ.gəl/ Explain

Pascal's triangle is a simplified version of the binomial theorem. Pascal's triangle starts at row 1 with the number 1. The second row contains two numbers: 1 and 1. In each iteration, adjacent values are added together to make the number for the next iteration.

Yang Hui triangle published in 1303 by Zhu Shijie.
Figure 2: Yang Hui triangle published in 1303 by Zhu Shijie. Click on the image to see a larger version.

Pascal's triangle in the western world is named for the French mathematician Blaise Pascal. However, it was studied earlier in India, Persia, China, and Italy. See figure 2.

Properties of Pascal's Triangle

  • The sum of the numbers of each row is twice that of the previous row.
  • The first number after the 1 in each row evenly divides all the other numbers if and only if the first number after the 1 is a prime number.
  • The 'shallow diagonals' of Pascal's triangle add up to the Fibonacci numbers. See figure 3.
  • The equation for Pascal's triangle is
    a_{nr}\\;\\equiv{\\;\\frac{n!}{r!\\left(n-r\\right)!}}\\;\\equiv{\\;{n \\choose r}}
    where n is the row number and r is the column in the row.

Pascal's triangle with the shallow diagonals added to make the Fibonacci numbers.
Figure 3: Shallow diagonals of Pascal's Triangle


  1. McAdams, David E.. All Math Words Dictionary, Pascal's triangle. 2nd Classroom edition 20150108-4799968. pg 135. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. pascal's triangle. Encyclopedia Britannica. Merriam-Webster. Last Accessed 12/3/2018.'s triangle. Buy the book

Cite this article as:

McAdams, David E. Pascal's Triangle. 4/27/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

4/27/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/1/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
10/25/2008: Initial version. (McAdams, David E.)

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