Pascal's Triangle

Definition

Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. It is named after the 17th-century French mathematician Blaise Pascal, who studied its properties extensively.

5 Must Know Facts For Your Next Test

1. The first row of Pascal's Triangle consists of a single 1, and each subsequent row begins and ends with a 1, with the intermediate numbers being the sum of the two numbers directly above them.
2. The numbers in Pascal's Triangle are symmetric, meaning that the numbers on the left side of the triangle are the same as the numbers on the right side, just in reverse order.
3. The sum of the numbers in each row of Pascal's Triangle is a power of 2, with the first row having a sum of 1 (2^0), the second row having a sum of 2 (2^1), and so on.
4. Pascal's Triangle has many applications in mathematics, including probability theory, combinatorics, and the expansion of binomial expressions.
5. The pattern of numbers in Pascal's Triangle can be used to generate other mathematical sequences, such as the Fibonacci sequence and the powers of 11.

Review Questions

• Explain how the numbers in Pascal's Triangle are generated and the relationship between adjacent numbers.
  • The numbers in Pascal's Triangle are generated by starting with a 1 in the first row, and then each subsequent number is the sum of the two numbers directly above it. For example, the third row of the triangle is 1, 2, 1, where the 2 is the sum of the 1's above it. This pattern continues, with each number being the sum of the two numbers directly above it, creating a triangular array of numbers.
• Describe the relationship between the numbers in Pascal's Triangle and binomial coefficients.
  • The numbers in Pascal's Triangle are also known as binomial coefficients, which represent the coefficients of the terms in the expansion of a binomial expression. For example, the binomial expansion of $(a + b)^n$ can be written as $\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$, where the coefficients $\binom{n}{k}$ are the numbers in the $n$th row and $k$th column of Pascal's Triangle.
• Analyze the symmetry and other properties of the numbers in Pascal's Triangle, and explain how these properties can be used to solve problems related to the multiplication of polynomials.
  • The numbers in Pascal's Triangle exhibit a high degree of symmetry, with the numbers on the left side of the triangle being the same as the numbers on the right side, just in reverse order. This symmetry can be used to simplify the multiplication of polynomials, as the coefficients of the resulting polynomial can be easily determined from the coefficients of the original polynomials. Additionally, the fact that the sum of the numbers in each row of Pascal's Triangle is a power of 2 can be used to efficiently compute the expansion of binomial expressions.

Related terms

Binomial Coefficients: The numbers in Pascal's Triangle are also known as binomial coefficients, which represent the coefficients of the terms in the expansion of a binomial expression.

Combinatorics: Pascal's Triangle is closely related to the field of combinatorics, which is the study of counting and enumeration problems.

Factorial: The numbers in Pascal's Triangle can be expressed in terms of factorials, which represent the product of all positive integers less than or equal to a given number.