A triangular array of binomial coefficients, used to solve problems in probability, algebra, and combinatorial analysis.
Basic properties of Pascal's Triangle: The basic structure, properties, and pattern of Pascal's triangle, including row and column sums, symmetry, and triangularity.
Binomial coefficients: The relationship between Pascal's Triangle and binomial coefficients (combinations), including how they are used to compute the entries of the triangle.
Pascal's identity: The recursive formula, known as Pascal's identity, used to generate the next row of Pascal's Triangle using the previous row.
Combinatorial proofs: The use of combinatorial arguments to prove various identities in Pascal's Triangle, such as the Hockey Stick Theorem and Vandermonde's Identity.
Fibonacci numbers: The relationship between Pascal's Triangle and the Fibonacci sequence, including the appearance of Fibonacci numbers in the diagonal elements of the triangle.
Applications to probability: The use of Pascal's Triangle in probability theory, including the binomial distribution and the probability of winning a game of chance.
Applications to algebra: The use of Pascal's Triangle in algebraic expressions, including the expansion of powers of binomials using the binomial theorem.
Applications to geometry: The use of Pascal's Triangle in geometry, including the Pascal Line and various related theorems.
Pascal's Pyramid: The extension of Pascal's Triangle to higher dimensions, resulting in Pascal's Pyramid and its properties.
Generalizations: Generalizations of Pascal's Triangle, including the Stirling Triangle and the Lah numbers.
Classic Pascal's Triangle: A triangular array of numbers in which the numbers in each row are the coefficients of the terms in the expansion of (x + y)ⁿ, where n is the row number.
Symmetric Pascal's Triangle: A variation of the classic Pascal's Triangle in which each row is a palindrome (reads the same backwards and forwards).
Modulo n Pascal's Triangle: A Pascal's Triangle constructed by computing the coefficients modulo n, where n is a given integer.
Binomial Pascal's Triangle: A Pascal's Triangle in which each entry is the binomial coefficient (n choose k), where n is the row number and k is the column number.
Fibonacci Pascal's Triangle: A Pascal's Triangle in which each entry is the sum of the two entries above it, similar to the Fibonacci sequence.
Lucas Pascal's Triangle: A variation of the Fibonacci Pascal's Triangle in which the initial row is (1,2,1) instead of (1,1).
Complementary Pascal's Triangle: A Pascal's Triangle in which each entry is the complement of the entry in the corresponding position of the classic Pascal's Triangle.
Sierpinski Triangle: A triangle pattern in which the numbers in each row are either 0 or 1 based on the previous row's pattern, resulting in a fractal structure.
Euler's Polygonal Number Triangle: A Pascal's Triangle in which each entry is the number of ways to partition the row number into distinct polygonal numbers, such as triangular or square numbers.
Stirling Pascal's Triangle: A Pascal's Triangle in which each entry is a Stirling number of the second kind, representing the number of ways to partition n objects into k non-empty sets.