2n 2n 1/2n + 2 1, 2 n +1 n -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Prove the following identity combinatorially.
![This equation is an example of a binomial coefficient identity. It is written as:
\[
\binom{2n}{n} + \binom{2n}{n-1} = \frac{1}{2} \binom{2n+2}{n+1}
\]
Explanation:
- The left side consists of two binomial coefficients. The first term, \(\binom{2n}{n}\), represents the number of ways to choose \(n\) elements from a set of \(2n\) elements. The second term, \(\binom{2n}{n-1}\), represents the number of ways to choose \(n-1\) elements from the same set.
- The right side features \(\frac{1}{2} \binom{2n+2}{n+1}\). This involves taking half of the binomial coefficient \(\binom{2n+2}{n+1}\), which represents choosing \(n+1\) elements from \(2n+2\) elements.
This identity demonstrates a relationship between these binomial coefficients, showing how combinations from different sets relate to one another.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd8afda51-bd9d-45b3-9002-026f5c1755d0%2F9902e405-7923-442c-8a74-03f27cf364ee%2Fwp4bhy_processed.png&w=3840&q=75)
Transcribed Image Text:This equation is an example of a binomial coefficient identity. It is written as:
\[
\binom{2n}{n} + \binom{2n}{n-1} = \frac{1}{2} \binom{2n+2}{n+1}
\]
Explanation:
- The left side consists of two binomial coefficients. The first term, \(\binom{2n}{n}\), represents the number of ways to choose \(n\) elements from a set of \(2n\) elements. The second term, \(\binom{2n}{n-1}\), represents the number of ways to choose \(n-1\) elements from the same set.
- The right side features \(\frac{1}{2} \binom{2n+2}{n+1}\). This involves taking half of the binomial coefficient \(\binom{2n+2}{n+1}\), which represents choosing \(n+1\) elements from \(2n+2\) elements.
This identity demonstrates a relationship between these binomial coefficients, showing how combinations from different sets relate to one another.
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