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.
Click on figure 1 to see the algorithm for building a Pascal's triangle.
 Figure 1: Pascal's Triangle
Figure 1 contains the first eight iterations
of Pascal's triangle. Click on the image to see how the triangle is constructed.

 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
where n is the row number and r is the column in the row.

 Figure 3: Shallow diagonals of Pascal's Triangle 

References
 McAdams, David E.. All Math Words Dictionary, Pascal's triangle. 2nd Classroom edition 201501084799968. pg 135. Life is a Story Problem LLC. January 8, 2015. Buy the book
 pascal's triangle. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 12/3/2018. http://www.merriamwebster.com/dictionary/pascal's triangle. Buy the book
Cite this article as:
McAdams, David E. Pascal's Triangle. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/pascalstriangle.html.
Image Credits
Revision History
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.)