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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Combinatorial Identities and Proofs
#### a. Combinatorial Proof for the Identity
**Given Conditions:**
- \( m \geq 0 \)
- \( n \geq 0 \)
- \( r \geq 0 \)
**Identity to Prove:**
\[
\binom{m}{0} \binom{n}{r} + \binom{m}{1} \binom{n}{r-1} + \cdots + \binom{m}{r} \binom{n}{0} = \binom{m+n}{r}
\]
**Note:**
\[
\binom{a}{b} = 0, \text{ whenever } b > a
\]
#### b. Proof of Another Identity
**Identity to Prove:**
\[
\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \cdots + \binom{n}{n}^2 = \binom{2n}{n}
\]
**Hint:**
- Find appropriate values for \( m, n, \) and \( r \) in the identity from part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a51d15d-a9a3-449d-a2a2-e3bf35bb9007%2Fbdc50d22-7159-4c35-98a3-dd43c195b288%2Fllvvxyh_processed.png&w=3840&q=75)
Transcribed Image Text:### Combinatorial Identities and Proofs
#### a. Combinatorial Proof for the Identity
**Given Conditions:**
- \( m \geq 0 \)
- \( n \geq 0 \)
- \( r \geq 0 \)
**Identity to Prove:**
\[
\binom{m}{0} \binom{n}{r} + \binom{m}{1} \binom{n}{r-1} + \cdots + \binom{m}{r} \binom{n}{0} = \binom{m+n}{r}
\]
**Note:**
\[
\binom{a}{b} = 0, \text{ whenever } b > a
\]
#### b. Proof of Another Identity
**Identity to Prove:**
\[
\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \cdots + \binom{n}{n}^2 = \binom{2n}{n}
\]
**Hint:**
- Find appropriate values for \( m, n, \) and \( r \) in the identity from part (a).
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