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d) (²) = (^z¹) + (−1) for all integers n and k such that n ≥ k≥1.

d) (²) = (^z¹) + (−1) for all integers n and k such that n ≥ k≥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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### Combinatorial Identity Simplification **d)** \(\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}\) for all integers \(n\) and \(k\) such that \(n \geq k \geq 1\). **Explanation:** This equation represents a fundamental identity in combinatorics known as Pascal's Rule. It states that the binomial coefficient \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from \(n\) elements, can be expressed as the sum of the number of ways to choose \(k\) elements from \(n-1\) elements and the number of ways to choose \(k-1\) elements from \(n-1\) elements. ### Example: For \(n = 5\) and \(k = 2\): \[ \binom{5}{2} = \binom{4}{2} + \binom{4}{1} \] By calculating the individual terms: \[ \binom{5}{2} = 10, \quad \binom{4}{2} = 6, \quad \binom{4}{1} = 4 \] Thus, \[ 10 = 6 + 4 \] This confirms the identity as correct for this particular example.
Transcribed Image Text:### Combinatorial Identity Simplification **d)** \(\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}\) for all integers \(n\) and \(k\) such that \(n \geq k \geq 1\). **Explanation:** This equation represents a fundamental identity in combinatorics known as Pascal's Rule. It states that the binomial coefficient \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from \(n\) elements, can be expressed as the sum of the number of ways to choose \(k\) elements from \(n-1\) elements and the number of ways to choose \(k-1\) elements from \(n-1\) elements. ### Example: For \(n = 5\) and \(k = 2\): \[ \binom{5}{2} = \binom{4}{2} + \binom{4}{1} \] By calculating the individual terms: \[ \binom{5}{2} = 10, \quad \binom{4}{2} = 6, \quad \binom{4}{1} = 4 \] Thus, \[ 10 = 6 + 4 \] This confirms the identity as correct for this particular example.
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