Pascal's triangle

Learning Objectives Construct Pascal's triangle to any specified row. Identify and describe key numerical patterns within Pascal's triangle, such as the sums of rows and the hockey-stick identity. Relate the entries in Pascal's triangle to combinatorial coefficients, C(n, k). Use Pascal's triangle to efficiently determine the coefficients for the expansion of a binomial of the form (a + b)^n. Apply the Binomial Theorem to fully expand polynomial expressions. Find a specific term in a binomial expansion without calculating the entire expansion. Ever tried to calculate (x + y)⁷ by hand? 🤯 There's a beautiful shortcut hidden in a simple triangle of numbers! This tutorial introduces Pascal's triangle, a powerful visual tool that reveals fascinatin...

TermDefinitionExample Pascal's TriangleA triangular array of numbers where each number is the sum of the two numbers directly above it. The edges of the triangle are all 1s.Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 (The '2' in Row 2 is the sum of the '1' and '1' from Row 1 above it). Binomial ExpansionThe process of raising a binomial (a two-term polynomial) to a power and expressing it as a sum of terms.The expansion of (x + y)² is x² + 2xy + y². CoefficientThe numerical factor of a term in a polynomial.In the term 6x², the coefficient is 6. Combination (nCr)The number of ways to choose 'r' items from a set of 'n' distinct items, where the order of selection does not matter. The values in Pascal's triangle correspond...

Construction Rule Each entry is the sum of the two entries directly above it. The outside entries of each row are always 1. Use this rule to build the triangle row by row. To find an entry, look to the row above and add the numbers that are diagonally to its left and right. Combinatorial Formula for Entries The entry in the n-th row and k-th position is given by the combination formula: C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} This formula allows you to calculate any entry in the triangle without constructing all the rows above it. Remember that n is the row number and k is the term position, both starting from 0. Binomial Theorem Pattern (a+b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{n}a^0 b^n This is the formul...