k n = ( 2 ) · ( 5 ) = ( 7 ) · (2 – 7 ) for k k-r Prove Newton's Identity: for n ≥k≥r≥ 0.....
k n = ( 2 ) · ( 5 ) = ( 7 ) · (2 – 7 ) for k k-r Prove Newton's Identity: for n ≥k≥r≥ 0.....
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 9: Prove Newton's Identity**
Prove the identity:
\[
\binom{n}{k} \cdot \binom{k}{r} = \binom{n}{r} \cdot \binom{n-r}{k-r}
\]
where \( n \geq k \geq r \geq 0 \).
---
**Explanation:**
- **\(\binom{n}{k}\):** This notation represents a binomial coefficient, defined as the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order. It is calculated using the formula \(\frac{n!}{k!(n-k)!}\).
- **\(\binom{k}{r}\):** Similarly, this represents the number of ways to choose \( r \) elements from a set of \( k \) elements.
- The identity states that the product of these two binomial coefficients is equal to the product of \(\binom{n}{r}\) and \(\binom{n-r}{k-r}\).
The conditions \( n \geq k \geq r \geq 0 \) ensure that all binomial coefficients are well-defined (non-negative integers). Proving this identity involves combinatorial arguments or algebraic manipulations using properties of factorials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc45dfd55-2702-405b-9288-88cd78d06c07%2F10129698-9dd9-4ccb-8ef2-6d29e848f931%2F0bwfuco_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 9: Prove Newton's Identity**
Prove the identity:
\[
\binom{n}{k} \cdot \binom{k}{r} = \binom{n}{r} \cdot \binom{n-r}{k-r}
\]
where \( n \geq k \geq r \geq 0 \).
---
**Explanation:**
- **\(\binom{n}{k}\):** This notation represents a binomial coefficient, defined as the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order. It is calculated using the formula \(\frac{n!}{k!(n-k)!}\).
- **\(\binom{k}{r}\):** Similarly, this represents the number of ways to choose \( r \) elements from a set of \( k \) elements.
- The identity states that the product of these two binomial coefficients is equal to the product of \(\binom{n}{r}\) and \(\binom{n-r}{k-r}\).
The conditions \( n \geq k \geq r \geq 0 \) ensure that all binomial coefficients are well-defined (non-negative integers). Proving this identity involves combinatorial arguments or algebraic manipulations using properties of factorials.
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