The Pascal Triangle is an arrangement of numbers in
the form of a triangle in such a manner that each number on the triangle is
equal to the sum of the two numbers above it. This triangle is named after
Blaise Pascal (16231662), a French mathematician who studied it extensively.
However, other mathematicians in Persia and China may have studied it centuries
before him.
Below is the Pascal Triangle:












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 










1 
7 
21 
35 
35 
21 
7 
1 




where the first eight rows are shown. The triangle
can be of enumerated for any size and may go into infinity.
It can be noted that the upper left and
upper right sides of the triangle are composed of the numeral 1. The other
elements of the triangle are constructed by placing into the respective
positions of the triangle a number equal to the sum of the two numbers above it.
One usefulness of the Pascal Triangle can
be seen on the coefficients of the binomial expansion of the term
(x + y)^{n}
For example, for n=5, the binomial
expansion of (x + y)^{5} is:
(1)x^{5} + (5)x^{4}y^{1}
+ (10)x^{3}y^{2} + (10)x^{2}y^{3} + (5)x^{1}y^{4}
+ (1)y^{5 }
where it can be noted that the
coefficients of the expansion corresponds to the elements of a row in the Pascal
Triangle. In this case, this is the 6^{th} row of the Pascal
Triangle.
More
Mathematical Recreations
