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and c) [8]3. Begins as 6) Compute a) Pascal triangle. Use it to find b) () 3 4 6 1 1 14 1 1

and c) [8]3. Begins as 6) Compute a) Pascal triangle. Use it to find b) () 3 4 6 1 1 14 1 1

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Pascal's Triangle and Its Applications #### Problem Statement 6) Compute: - a) Pascal's triangle. - Use it to find: b) \( \binom{8}{4} \) c) \( [8]_3 \). #### Explanation Pascal's triangle is a triangular array of numbers, where each number is the sum of the two numbers directly above it in the previous row. It starts with a 1 at the top as follows: ``` 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ``` - **First Row**: \(1\) - **Second Row**: \(1, 1\) - **Third Row**: \(1, 2, 1\) - **Fourth Row**: \(1, 3, 3, 1\) - **Fifth Row**: \(1, 4, 6, 4, 1\) #### Applications **b) Find \( \binom{8}{4} \):** To find the value of \( \binom{8}{4} \) using Pascal's triangle, you would continue building the triangle up to the 9th row (remember, rows are 0-indexed in Pascal's Triangle) and then select the 5th element in that row (also 0-indexed). **c) Find \( [8]_3 \):** The notation \( [8]_3 \) typically refers to a coefficient in a particular polynomial expansion, but further context is needed to specify exactly what this expression refers to (e.g., a binomial expansion or another polynomial). In many contexts, involving Pascal's triangle, it's related to combinations or specific uses in algebraic expansions. Building Pascal’s triangle helps visualize binomial coefficients, which are integral to combinatorics and various branches of mathematics, such as probability and algebra.
Transcribed Image Text:### Pascal's Triangle and Its Applications #### Problem Statement 6) Compute: - a) Pascal's triangle. - Use it to find: b) \( \binom{8}{4} \) c) \( [8]_3 \). #### Explanation Pascal's triangle is a triangular array of numbers, where each number is the sum of the two numbers directly above it in the previous row. It starts with a 1 at the top as follows: ``` 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ``` - **First Row**: \(1\) - **Second Row**: \(1, 1\) - **Third Row**: \(1, 2, 1\) - **Fourth Row**: \(1, 3, 3, 1\) - **Fifth Row**: \(1, 4, 6, 4, 1\) #### Applications **b) Find \( \binom{8}{4} \):** To find the value of \( \binom{8}{4} \) using Pascal's triangle, you would continue building the triangle up to the 9th row (remember, rows are 0-indexed in Pascal's Triangle) and then select the 5th element in that row (also 0-indexed). **c) Find \( [8]_3 \):** The notation \( [8]_3 \) typically refers to a coefficient in a particular polynomial expansion, but further context is needed to specify exactly what this expression refers to (e.g., a binomial expansion or another polynomial). In many contexts, involving Pascal's triangle, it's related to combinations or specific uses in algebraic expansions. Building Pascal’s triangle helps visualize binomial coefficients, which are integral to combinatorics and various branches of mathematics, such as probability and algebra.
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