Pascal's Triangle

Pascal's triangle was also known as the figurate triangle, the combinatorial triangle, and the binomial triangle. The triangle was first given the name, "Pascal's triangle," by a mathematician named Montmort in 1708. Montmort wrote the numbers in the form below known as the combinatorial triangle.

The combination of numbers that form Pascal's triangle were well known before Pascal, but he was the first one to organize all the information together in his treatise, "The Arithmetical Triangle." The numbers originally arose from Hindu studies of combinatorics and binomial numbers, and the Greek's study of figurate numbers. The Chinese also wrote about the binomial numbers in "Precious Mirror of the Four Elements" in 1303. The figurate numbers were known over 500 years before Christ. There are square and triangular figurate numbers. The first four of each are shown below.

The triangular numbers:

The square numbers:

Additional square and triangular numbers are formed by increasing the size of each respectively. Actually, figurate numbers can be formed from any polygon. Another set of figurate numbers could be formed using the pentagon, a polygon with five sides. The figurate numbers were studied heavily to learn about counting numbers and arrangements. For example, if a woman was asked to determine which of two sacks of gold coins was worth more, she would probably have to count the coins. To count the coins, the best approach would be to stack the coins into short stacks of a given number. Then the number of stacks could be counted. Counting numbers, looking at the patterns and studying the ways objects could be arranged led to the numbers in Pascal's triangle. The study of combining or arranging objects by various rules to create new arrangements of objects is called Combinatorics, an important branch of mathematics. Pascal's triangle in its current form is shown below. It is the same as the above combinatorial triangle rotated 45 degrees clockwise.

Each new row in Pascal's triangle is solved by taking the top two numbers and adding them together to get the number below.

The triangle always starts with the number one and has ones on the outside. Another way to calculate the numbers is Pascal's Triangle is to calculate the binomial coefficients, written C(r;c). A formula for the binomial coefficients is r! divided by c! × (r-c)!. The variable r represents the row and c, the column, of Pascal's Triangle. The exclamation point represents the factorial. The factorial of a number is that number times every integer number less than it until the number one is reached. So 4! would be equal to 4 × 3 × 2 × 1 or 24.