Pascal’s (Zhu Shijie’s) Triangle
Pascal’s Triangle is a special triangular arrangement of numbers used in many areas of mathematics. It is named after the famous $17$ ^{ th } century French mathematician Blaise Pascal because he developed so many of the triangle’s properties. However, this triangular arrangement of numbers was known by the Arabian poet and mathematician Omar Khayyam (c $1044$ - $1123$ ) and the Chinese mathematician Zhu Shijie (c $1260$ - $1320$ ) some $250$ years before Pascal.
$\begin{array}{cccccccccccccccc}& & & & & & & 1& & & & & & & & \text{Row0}\\ & & & & & & 1& & 1& & & & & & & \text{Row1}\\ & & & & & 1& & 2& & 1& & & & & & \text{Row2}\\ & & & & 1& & 3& & 3& & 1& & & & & \text{Row3}\\ & & & 1& & 4& & 6& & 4& & 1& & & & \text{Row4}\\ & & 1& & 5& & 10& & 10& & 5& & 1& & & \text{Row5}\\ & 1& & 6& & 15& & 20& & 15& & 6& & 1& & \text{Row6}\\ 1& & 7& & 21& & 35& & 35& & 21& & 7& & 1& \text{Row}7\end{array}$
At the top of the triangle is a $1$ , which makes up the $0$ ^{ th } row. The $1$ ^{ st } row $\left(1,1\right)$ contains two $1$ s each formed by adding the two numbers above them, one to the left and one to the right, in this case $0$ and $1$ . (All numbers outside the triangle are $0$ s.) Do the same to create the $2$ ^{ nd } row; $0+1=1,1+1=2,1+0=1$ and all subsequent rows.
A number in the triangle can be found by using ${}_{n}C{}_{r}$ ( $n$ choose $r$ ), where $n$ is the number of the row and $r$ is the number of the element in that row. $\left({}_{n}C{}_{r}=\frac{n!}{r!\left(n-r\right)!}\right)$ This is especially helpful to find a particular term in the expansion of a binomial in the form ${\left(x+y\right)}^{n}$ .
Example:
Find the $4$ ^{ th } term in the $6$ ^{ th } row of the triangle.
${}_{6}C{}_{4}=\frac{6!}{4!\left(6-4\right)!}=\frac{6!}{4!2!}=15$
(Remember: the first $1$ in each row is the $0$ ^{ th } element so this is correct.)
Sum of rows: The sum of the numbers in any row is equal to ${2}^{n}$ , when $n$ is the number of the row.
$\begin{array}{l}{2}^{0}=1=1\\ {2}^{1}=2=1+1\\ {2}^{2}=4=1+2+1\\ {2}^{3}=8=1+3+3+1\\ {2}^{4}=16=1+4+6+4+1\end{array}$
and so forth.
Prime numbers: If the first element in a row is a prime number (remember the first 1 in any row is the $0$ ^{ th } element.) all of the numbers in that row (excluding the $1$ s) are divisible by it.
For example in the $7$ ^{ th } row ( $1,7,21,35,35,21,7,1$ ) $7,21,35$ are divisible by $7$ .
In Algebra, each row in Pascal’s Triangle contains the coefficients of the binomial $\left(x+y\right)$ raised to the power of the row.
$\begin{array}{l}{\left(x+y\right)}^{0}=1\\ {\left(x+y\right)}^{1}=1x+1y\\ {\left(x+y\right)}^{2}=1{x}^{2}+2xy+1{y}^{2}\\ {\left(x+y\right)}^{3}=1{x}^{3}+3{x}^{2}y+3x{y}^{2}+1{y}^{3}\\ {\left(x+y\right)}^{4}=1{x}^{4}+4{x}^{3}y+6{x}^{2}{y}^{2}+4x{y}^{3}+1{y}^{4}\end{array}$
and so forth.
Another major area where Pascal’s Triangle shows up and is very useful is in probability where it can be used to find combinations .
Interesting Number Patterns:
Many interesting number patterns can be found in the triangle. Included are the Fibonacci sequence , Triangular and Square Numbers (found in the diagonals starting with row $3$ .) and Polygonal Numbers.
Another interesting connection is to Sierpinski’s Triangle. When all of the odd numbers in Pascal’s Triangle are filled in and the evens are left blank, the recursive Sierpinski Triangle fractal is revealed.
Each of these are fascinating topics which warrant further research on your part.