Pascal's triangle

Pascal's Triangle is a triangular array of numbers that has found applications in various fields such as mathematics, statistics, computer science, and engineering. Each number in the triangle is the sum of the two numbers directly above it. The triangle is named after Blaise Pascal, a French mathematician, physicist, and philosopher, who made significant contributions to its study in the 17th century, although the triangle had been known and used by mathematicians in India, Persia, China, and Italy centuries before Pascal.

Construction

Pascal's Triangle is constructed starting with a single 1 at the top, known as the apex. Each subsequent row is created by adding the number above and to the left with the number above and to the right, treating empty spaces as 0. The edges of the triangle are always 1s.

Example

The first six rows of Pascal's Triangle look like this:

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Properties

Pascal's Triangle has several remarkable properties and patterns, including:

• Symmetry: The triangle is symmetric along its vertical axis.
• Binomial Coefficients: Each number in the triangle represents a binomial coefficient, which is crucial in the binomial expansion theorem.
• Fibonacci Sequence: The sum of the numbers in the diagonals of Pascal's Triangle gives the Fibonacci sequence.
• Prime Numbers: The properties of the rows can reveal interesting patterns related to prime numbers.

Applications

Pascal's Triangle has wide-ranging applications in various fields:

• In mathematics, it is used in combinatorics to find combinations and permutations.
• In probability theory, it helps in understanding the binomial distribution.
• In computer science, it is used in algorithms and computing binomial coefficients efficiently.

See Also

• Binomial Theorem
• Combinatorics
• Fibonacci Sequence